The common experience of turning on a light switch certainly shows that light travels very quickly. But careful experiments reveal that it travels at a finite speed. This speed, which we call "c," is measured to be 300,000,000 meters per second.
The speed of light is strange in that it has the same value independent of the relative velocity between the source and the observer. This fact is an experimental one that can only make sense if relative motion changes the relationship between space and time intervals to keep the distance covered by light per unit time the same for all observers.
The fact that space and time must get mixed up to keep the speed of light constant implies that, in some sense, space and time must be the same, despite our habit of measuring space in meters and time in seconds. But if time and space are similar to the extent that they can be converted one into the other, then one needs some quantity to convert the units--namely, something measured in meters per second that can be used to multiply seconds of time to get meters of space. That something, the universal conversion factor, is the speed of light. The reason that it is limited is simply the fact that a finite amount of space is equivalent to a finite amount of time.
Another explanation of light's finite nature can be obtained from thinking about what we mean by light itself. Light, by definition, is an electromagnetic wave, a propagating disturbance in space and time that carries information about the acceleration of charges.
Were there an infinite value for the speed of light, light itself would not exist at all. Mathematically, the wave equation that describes light as an electromagnetic wave would lose its time-dependence.
In physical terms, an electromagnetic wave arises due to the finite time it takes for news of the change of location of an accelerated charge to arrive at a distant point. Think of an electric charge as being like a hedgehog with flexible rubber spikes going out to infinity in all directions. These spikes represent the electric field lines, the lines along which a test charge would move.
If the charge is jerked, the segments of the spikes close to the charge will move, but those farther out will still point in their original directions. The result is that each spike will get a kink that moves out to infinity. This kink relays the news that the charge has moved to the distant parts of the spikes and corresponds to an electromagnetic wave. If the wave moves infinitely fast, it is as if it were not there at all; the spikes are infinitely stiff and the news gets out to everywhere without any seeming kinks. In other words, there would be no electromagnetic wave, and thus no light.
The previous two arguments are two slightly different ways to say that if you think light is a wave, then it has to be something that propagates and takes time to go from one point to another. In other words, it has to travel at a finite speed. Infinite speed of propagation is an instantaneous magical change in things everywhere all at once, and not a wave at all!