If a photon has no mass why doesn't it have infinite speed? Please help a naïve layperson understand -- if a photon has no mass, why is its velocity limited at all? Shouldn't a particle with no mass be able to travel at an infinite speed?
 A: That light moves with a fixed speed in vacuum, in all reference systems is an experimental fact. Maxwell's equations fit so well all macroscopic electromagnetic data that the speed of light is fixed is not under question. It is inherent in the construction of the classical theory.
Light is made up by a zillion of photons. Photons are elementary particles in the standard model of particle physics. Elementary particles in motion with high velocities do not obey Galilean tranformations, which are implied in your question. These are true for macroscopic objects and velocities far away from the velocity of light. For the microcosm of elementary particles it is Lorenz transformations which define the velocities and the limits for transformations from one system to the other.
In Lorenz transformations the maximum velocity achievable is the velocity of light, by all zero mass particles. Particles with mass can never reach it.
Thus, as the smallest particle of light is the photon, the  photon must have zero mass since light's speed is limited by c.
A: There is no way to explain this without explaining relativity first. In Galilean universe (the "classical" physics, which is what most people intuitively assume and think about), light speed cannot be explained. Indeed, the Maxwell equations which is describe how light works, were the first clue that our understanding of space-time was flawed.
So the real question is: how to explain special relativity to a layperson. There are many approaches taken by documentaries and kids' books... I'd just say the following:
Velocity (as in, distance traveled, divided by time taken, measured both by the same person) is limited because the nature of our universe is such, that space and time are geometrically connected, similar to saying "why the largest possible angle in triangle is 180 degrees" or "why there's no north at the north pole". Forces cause acceleration and change momentum, but at large velocities, increasing momentum doesn't change the velocity all that much anymore.
There's sort of an analogy from real life... consider places A and B, and you want to get there in the shortest route possible. As long as people take very long and twisted ways around (the slow ways), then you can always find a better route and you don't even notice any limit (like our ancestors didn't need to care about the light speed)... but once you get to the really fast ways, you start realizing, there is a shortest way: the straight line. There's no shorter path than that, no matter how hard you try. It's pure geometry. There's no magical "speed limit" in the universe, we're just measuring a quantity that mathematically can't be larger than the speed of light. If you measure momentum, it can be as large as it likes.
Here's where the photons come in: in relativity, the relationship between momentum $p$, energy $E$ and velocity $v$ is $v=pc^2/E$. There's also the relation $E^2=(mc^2)^2+(pc)^2$. So, if you have mass, the energy will always be larger than $pc$, and the velocity will always be smaller than $c$. But for light, it's always $E=pc$, so $v=c$ (and can't be any different).
EDIT: how come the light speed isn't infinite? Because if it were, our universe would be much different in ways you'd most certainly notice. For wave equation to make sense, the wave speed has to be finite, so the fact that electromagnetic waves exist (Maxwell equations hold) proves it must be finite. If not, we'd have no induction, no radio, no interference.... it's quite natural for us to assume that changes in the environment take time to get noticed somewhere else, so much, that people struggled for a long time how gravity and such work (forces at a distance were problematic, and only resolved when we realized there are fields that carry this information through space). In that respect, finite speed of light made some things easier to understand. Instantaneous action at a distance would break a lot of physics.
A: In the special relativity, as explained by Einstein, there are two possibilities. Either the speed of any particle is limited at all or not. Now if the speed is not limited, the Galilean theory pops out. But as we already know, that does not properly explain the transformation of velocities from one to another reference frame accurately when dealing with high velocities. So only other case is that the velocity is limited, if so what is it?? It comes out that this velocity is precisely the velocity of light. 
So, any particle which is mass less travels at the speed c, which is why the speed of light is also c.
A: The speed limit c for massless particles is a characteristic of space.
In accordance with Maxwell's equation, speed of light which is the speed of massless particles is reflecting the characteristics of vacuum 


*

*vacuum permittivity $\mu_0$ and  

*vacuum permeability $\epsilon_0$.


$$ c = \frac {1}{\sqrt {\mu_0 \epsilon_0}}$$
Such a speed limit of Maxwell's equations is incompatible with Galileis relativity and led 1905 to Einstein's special relativity.
