Isn't the accelerating expansion of the universe intuitive? I have a question about accelerating expansion of the universe.  My understanding is that Hubble said the farther our we look into the universe, the higher the redshift, therefore the faster things are moving.  
But my question is, why is this surprising?  If the universal expansion works like every other explosion (expanding quickly at first, then slowing as it grows), wouldn't this be exactly what we expect to see?  I know I must be missing something as there are many people much smarter than myself working on it, but here's my reasoning, and I would appreciate some help:
Consider 3 arbitrary times in the history of the Universe $t_0$, $t_1$, $t_2$, where $t_0$ is shortly after the big bang, $t_1$ in the middle, and $t_2$ close to the present.  Shortly after $t_0$ the universe is moving quickly, slower at $t_1$ and close to current rate at $t_2$.
If it's true that the farther out in space we look, the farther back in time we look, wouldn't $t_0$ be very close to the edge of the observable Universe, $t_1$ slower, and closer in distance to use, with $t_2$ being the closest?
 A: First you should change your cartoon picture of the Big bang (an exploding egg of matter and energy in empty space). In Big bang theory the picture of evolution of Universe starts with the Universe(space) already expanding without any description of what started this expansion. Its all about the aftermath of the bang i.e. How the Universe evolved with time.
What Hubble did found with the observation of Cepheid variables and the red shift of the distant Galaxies was that farther the Galaxy was faster it seemed to move away from us. And this data gave an approximate linear relationship between the relative velocity of galaxies and the distance they were at from us.

Its same situation as the observers will find if they were on a stretching rubber band.
Let there be three observers (a,b,c) on a straight line (equally spaced) as above in the figure. When the rubber is stretched by a constant rate, the relative velocity of c with respect to a is two times that of b. . Twice the relative distance twice is the relative velocity , as the distance between a and b and b and c equally expanded in same time and thus the ratio $v/d$ (relative velocity/relative distance) is  constant and similar is what the picture (but evenly in all directions) of the Universe as the empirical relation  of Hubble shows- 
i.e.  $v/d=H$ where $H$ is called Hubble's constant.
So we saw that the Universe is expanding by which we mean the space between two points is expanding with time. Friedmann made a mathematical model of expanding Universe using the general theory of relativity which explains kind of expansion we see today, if we trace back the picture of the Universe to early times (according to friedmann equations) it results into a state of the Universe where,matter and energy density is very high. And from this state the Universe was set expanding to evolve to the state we see it today (this is the real bang in the Big bang which the big bang theory didn't give any description about). 
Theory models the evolution of matter and energy from hot dense soup of particles to the state we see it today in. As I mentioned its the aftermath of a bang. It's the FLRW (FRW) metric that help us understand the expansion with the time.
Astronomers saw something amazing from the observations of the redshifts of Type Ia supernova that the rate at which the Universe is expanding is increasing with time. Here 'expansion rate increases or expansion accelerates means the rate at which the distance between two points is increasing is increasing with time.' What I mean by this can be visualized with the same example of the stretching rubber band. 
In this case the rate at which the rubber band is stretched is increasing with time. Which mean the rate at which the distance between two observers is increasing is increasing with time.
And this is what we relate to an accelerating Universe (whereas our prior understanding of expansion tell us that the rate of expansion should decrease with time). There's a possibility for it (acceleration) may be happening due to the repulsive gravity because of Dark energy, which possibly can be  energy of Vaccum. Observations too fit into what this picture predicts, as John in his answer mentioned that People say it was expanding fast at $t_0$, slower at $t_1$ and faster again at $t_2$ (where $t_0$,$t_1$ and $t_2$ belong to some specific time period in the history of Universe as observations say.) 
A: 
My understanding is that Hubble said the farther our we look into the Universe, the higher the redshift, therefore the faster things are moving. 

Pretty much. If you were some raisin in a rising cake, you'd say the same of the other raisins.   

But my question is, why is this surprising? 

It isn't. What was surprising is that the expansion rate is increasing. See the accelerating universe on Wikipedia. Perlmutter Schmidt and Riess shared a Nobel prize for it.  

If the Universal expansion works like every other explosion (expanding quickly at first, then slowing as it grows), wouldn't this be exactly what we expect to see? 

No. What we're seeing is something more like drooping silly-putty or a bubble-gum balloon in a vacuum. The expansion isn't slowing down. It's speeding up. 

Consider 3 arbitrary times in the history of the Universe t0, t1, t2, where t0 is shortly after the big bang, t1 in the middle, and t2 close to the present. Shortly after t0 the Universe is moving quickly, slower at t1 and close to current rate at t2.

People say it was expanding fast at t0, slower at t1, and faster again at t2. This is why the expanding universe is depicted with a flared wine flute shape. Like this:

Public domain image courtesy of NASA, see Wikipedia

If it's true that the farther out in space we look, the farther back in time we look

Yes, that's right. 

wouldn't t0 be very close to the edge of the observable Universe, t1 slower, and closer in distance to use, with t2 being the closest?

Kind of. Check out the Hubble ultra deep field. 
