Why is the observable universe so big? The observable universe is approximately 13.7 billion years old. But yet it is 80 billion light years across. Isn't this a contradiction?
 A: The universe is commonly defined as the totality of everything that exists, including all physical matter and energy, the planets, stars, galaxies, and the contents of intergalactic space. 
No one knows if the universe is infinitely large, or even if ours is the only universe there is.
Although our view of the universe is limited, our imaginations are not. Astronomers have indirect evidence that the universe of galaxies extends far beyond the region we can see. But no one knows if the whole universe is infinitely large - large beyond limit.
According to the leading theories, other parts of the universe may look very different from our own - and may even have different laws of nature. We may never be able to find out for sure. But it is possible that clues to the answer lie in plain view, just waiting to be discovered!
I should note also the "80 billion light years across" it doesn't count as a contradiction. I don't know what your reference is but I believe that this concerns the region that we can see yet of this Universe.   
A: You can travel as close to the speed of light as you want, but (assuming you're made from matter) you can't travel at the speed of light or faster.
So there's no reason why galaxies can't be receding from each other at 99.9999% the speed of light.  However, this isn't the whole story, as the expansion of spacetime can do funny things.  The universe isn't actually expanding as you'd expect from everyday life, as a explosion would, and there wasn't a "single point" that you can point to and say "that's where the big bang happened", it happened where you're sitting, and on the dark side of the moon, and just around the corner from Alpha Centauri...everywhere in fact.  
Picture a graph - the point on the graph aren't moving away from each other (ie point A goes from (1,1) to (2,2) and point B goes from (5,5 to 7,7)) - instead, the whole graph, piece of paper and all, is being stretched.
This stretching is allowed to happen faster than the speed of light, as no matter actually has to change coordinates for it to happen.
In relation to the title of your question, the best answer is to ask "compared to what?".
A: This question implicitly refers to the visible universe, but we should state that explicitly, as otherwise the question doesn't make any sense.
It may seem like we shouldn't be able to see more than 13.7 billion light-years (13.7 giga-light-years, or glyrs) away, but that reasoning omits the expansion of spacetime according to General Relativity. A photon emitted from somewhere near the beginning of the Universe would have traveled nearly 13.7 glyrs if you had measured each light-year just as the photon crossed it, but since those light-years that you measured have expanded since the photon passed through, that distance now adds up to about 80 glyrs.
A: The radius of the observable universe is about 46 billion light years, which is considerably greater than its age of about 14 billion years. Since the radius of the observable universe is defined by the greatest distance from which light would have had time to reach us since the Big Bang, you might think that it would lie at a distance of only 14 billion light years, since $x=ct$ for motion at a constant velocity $c$. However, a relation like $x=ct$ is only valid in special relativity. When we write down such a relation, we imagine a Cartesian coordinate system $(t,x,y,z)$, which in Newtonian mechanics would be associated with a particular observer's frame of reference. In general relativity, the counterpart of this would be a Minkowski coordinate frame, but such frames only exist locally. It is not possible to make a single frame of reference that encompasses both our galaxy and a cosmologically distant galaxy. General relativity is able to describe cosmology using cosmological models, and this description is successful in matching up with observations to a high level of precision. In particular, no objects are observed whose apparent ages are inconsistent with their distances from us.
One way of describing this difference between special relativity's $x=ct$ and the actual distance-time relationship is that we can think of the space between the galaxies as expanding. In this verbal description, we can imagine that as a ray of light travels from galaxy A to galaxy B, extra space is being created in between A and B, so that by the time the light arrives, the distance is greater than $ct$.
None of this has anything to do with inflation. Inflation makes certain testable predictions about cosmological observations (e.g., it predicts that the universe is spatially flat), but it's irrelevant for understanding why the radius of the observable universe has the size it does in comparison to the age of the universe. Inflation may not even be correct. If inflation turns out never to have happened, it will have no effect on this particular question.
It turns out that we can get a surprisingly good estimate of the size of the observable universe using a simplified FRW cosmological model consisting only of dust, i.e., nonrelativistic matter. The approximation is good because the universe has spent most of its history dominated by matter, with only a very short period early on that was radiation-dominated, and another fairly recent era that is dominated by the cosmological constant. In accord with the current observational data, we make a second approximation, which is that the universe is spatially flat. In a spatially flat FRW model, the $r-t$ part of the metric is of the form $ds^2=dt^2-a^2dr^2$, where the scale function $a$ depends on time. For a photon, $ds=0$, and we can then show that the proper distance traversed by a photon since shortly after the Big Bang is given by $L=a \int dt/a$. For a matter-dominated solution, $a$ is proportional to $t^{2/3}$, and we find $L=3t$. This is quite close to the $L/t$ ratio of about 3.3 given by the most realistic models. It also makes sense that the result is somewhat greater than 3, because the universe has now entered an era in which its expansion is accelerating. In the future, $L/t$ will become greater and greater.
A: This is not a contraction.  Special Relativity does not say that nothing can travel faster than the speed of light.  Rather, it says that no regular matter can travel faster than the speed of light.  Therefore, instead of thinking galaxies move away from each other, think that the space between them is increasing.  That would mean that the space itself can beat the speed of light.
A: Briefly...
Consider a photon emitted 13.8 Gly ago (Gly = billion light years). As it travels through space, not only its getting farther away from its source due to traveling at the speed of light $c$, but photons distance also increases as the universe grows.

Technically...
To get technical, the expansion of the universe is captured by the scale factor $a(t)$ which is governed by the Friedmann's equation:
$$\frac{(a'(t)/a(t))^2}{H_ 0^2}=\frac{\Omega _{\text{R0}}}{a(t)^4}+\frac{\Omega _{\text{M0}}}{a(t)^3}+\frac{\Omega _{\text{$\kappa $0}}}{a(t)^2}+\Omega _{\text{$\Lambda
$0}}$$
where 
the Hubble constant $H_0 = 67.8 \frac{\text{km}/\text{s}}{\text{Mpc}} = 0.0693 /\text{Gyr}$, 
the present value of the radiation density $\Omega _{\text{R0}} = 0.0000905$, 
the present value of the matter density 
$\Omega _{\text{M0}} = 0.308$ (which mainly consist of dark matter) , 
the present value of the curvature density $\Omega _{\text{$\kappa $0}} = 1 - (\Omega _{\text{R0}} + \Omega _{\text{M0}} + \Omega _{\text{$\Lambda
$0}}) = 1$, 
and the present value of the cosmological constant $\Omega _{\text{$\Lambda $0}} = 0.692$. These values are from the Plank Collaboration 2015. This equation can only be solve numerically. The following figure shows scale factor as a function of time:

In cosmology, the amount of distance a photon can travel in a given time is known as the comoving distance or the comoving horizon denoted by $\eta$:
$$\eta = \int_{0}^{t}\frac{c dt}{a(t)}$$
which again has to be evaluated numerically. By numerically integrating from 13.799 Gyr ago until the present time, a particle horizon of 43.5 Glyr is obtained which is the radios of the observable universe. One should take this value with a grain of salt since it is highly sensitive to $H_0$.

Biologically...
I would like to focus on "Why the observable universe is so big?" from different perspective. Alternatively we can ask why the observable universe is 13.8 Gyr old! This is not such an odd question to ask if we state it as the following: "Why are we, as observers, happen to look at the universe 13.8 Gyr after the big bang?" Given that intelligent life on earth took ~3.5 Gyr to rise and the conditions of the universe were too harsh to permit evolution of life during the first ~10 Gyrs, one could expect that most pockets of intelligent observers would form and study the universe ~15 Gyr after the big bang. However, it is very surprising to find ourselves at the dawn of formation of life given that life would flourish for many giga-years to come. 
A: Yes this does sound like it might be a contradiction but in fact it isn't. It's because the universe has been expanding rapidly in every direction since the big bang and our observations are limited by the speed of light.
For example if we observe a distant quasar that appears to be 10 billion light years away, the light from the quasar is 10 billion years old (which is why quasars are known for being some of the most ancient phenomena in the universe). In the time it took for that light to reach us the universe has been expanding. In fact the expansion has been accelerating all that while so the distance between us and that quasar is at this present time considerably larger than 10 billion light years.
If we had a means to observe distant objects as they appear right at this instance, not only would we have a type of time machine but we might observe the universe to be 80 billion light years across, although I cannot vouch for the accuracy of the 80 billion light year figure given on the wikipedia page.
Astronomers and physicists also struggle with the question of "what is it that the universe expanding into". Is it expanding into empty space and if so is there anything that lies out there beyond our universe at some vast distance? If so, the universe may be infinite.
Or does the universe fold back in on itself on some higher dimensional plain i.e. if we had a hypothetical means to travel faster that the rate of the expansion of the universe and travelled in a straight line, would we eventually end up back in the same spot? In this scenario the universe would have a theoretical boundary at any one time.
A: The question asks about opposite edges of the observable universe, but it's a special case of a more general question:

How can two objects that move away from a common point for a time $Δt$ end up more than $2cΔt$ apart?

You don't need to understand general relativity to understand the answer to this question, since it can happen even in special relativity. For the rest of my answer I'll assume a basic familiarity with special relativity.
Alice and Bob zero their stopwatches at a common place and time, then move away from each other for 1 second as measured by the stopwatches (or as measured by saying "one potato"). How far apart are they at the end? That is, what is the spacetime interval between the two events of the stopwatches reading 1 second?
It's easy to show* that if they both move inertially, then the interval at the end is $\sqrt{2(γ{-}1)}$ light seconds, where $γ$ is the gamma factor of their relative speed. Since $γ$ can be arbitrarily large, so can the distance. It exceeds 2 light seconds when $γ>3$ ($v \gtrsim 0.94c)$, and it exceeds any finite value for a large enough (sublight) relative speed.
In Minkowski spacetime, the triangle inequality only holds when all three sides of the triangle are spacelike. In other cases, the length of the third side isn't constrained at all by the lengths of the other two. Our thought-experiment is the case where two of the sides are timelike and the third is spacelike. The case where all three sides are timelike is the twin paradox: the stay-at-home twin can be arbitrarily old when the traveling twin returns from their two-second jaunt.
We can make this look a little more like a cosmology by introducing Carol, Ted, etc. who also move inertially away from the same point (which we'll dub the big bang) for 1 proper second. They move at different speeds, but we'll require that the relative speeds of nearest neighbors be similar to each other (this is the homogeneity assumption in cosmology). Also, instead of measuring the straight-line spacetime interval between the most distant movers, we'll measure the intervals between nearest neighbors and add them up.
If you plot the worldlines of all the movers, the points where their stopwatches read 1 second will lie on a hyperbola with the equation $t^2-x^2 = 1\,\text{s}^2$ in inertial coordinates.** The distances we measure are polygonal approximations to distances along this hyperbola. You might think that these distances would be longer than the straight-line distances; actually, they're shorter, because as usual everything is backwards in spacetime. But they aren't enough shorter to change the fundamental conclusion that arbitrarily long distances are possible. You can always make a larger distance by increasing the relative speed of adjacent movers, or by adding more of them at the extremes.
This toy model is more than just an analogy. If you start with the ΛCDM model that describes the real world (post-inflation) and take all of the densities continuously to zero in an appropriate way, it continuously morphs into this toy model. In every intermediate model, sufficiently distant objects are farther apart than $2c$ times the cosmological time since the big bang, and there's no point in the process of taking the limit at which the reason for that changes. The spacetime of the real world is curved, but the nonzero curvature isn't the reason that this counterintuitive behavior is possible. The real reason is the mixed metric signature of spacetime.

Aside from the failure of the triangle inequality, another reason people find this so confusing is probably the heavy emphasis on Cartesian (inertial) coordinate systems when teaching special relativity. Einstein had a good reason in 1905 to construct this type of coordinate system: he was trying to make a connection to the traditional Newtonian picture of space and time for the benefit of his audience which still believed in that picture. Today, 115 years later, no one believes in that picture, but we still teach it. There's so much drilling of Cartesian coordinates and related formulas (time dilation, length contraction, etc.) that people get the idea that this is how the universe really works. In truth, the universe doesn't care about our coordinate systems. The closest thing in the real world to a universal network of synchronized clocks is the expanding galaxies, and they don't follow a Cartesian grid, but a kind of polar grid (albeit on a curved manifold). If you define "distance" and "time" by reference to a Cartesian grid, then you can't get more than $Δx=2cΔt$ away; but if you define them by reference to the polar grid that actually (after a fashion) exists, then you can – even in the limit where spacetime is flat.

* Pick units where 1 second = 1 light second = 1, and inertial coordinates where the starting point is $x=t=0$, Alice doesn't move, and Bob moves on the x axis. Then Alice's final position is $(x,t)=(0,1)$ and Bob's is $(x,t)=(γβ,γ)$, and the distance between them is $\sqrt{(γβ-0)^2-(γ-1)^2}$ = $\sqrt{2γ-2}$.
** If you add more spatial dimensions, it's a hyperboloid, which in Minkowski space is a surface of constant negative curvature. This model is a zero-density limit of the FLRW cosmology, and this is why space is negatively curved in that limit, and not flat as you might expect: the FLRW coordinates are analogous to polar coordinates and "space" is a sphere of constant radius (cosmological time). In a Euclidean space it would have constant positive curvature; in Minkowski spacetime everything is backwards.
