Good question! There are two answers, one short and one a bit longer but more true (this is something about science -- sometimes there are several layers of truth and some are better answers and can explain more things).
First off, you might have heard of E=mc. While understanding this equation is not what I want to get into, it basically says that mass (m) and energy (E) can be understood as the same thing. So if gravity can affect mass, it can affect energy, like light. If you want to know more about this famous equation, there is another question about it on ScienceLine.
The second, "better," answer is that gravity really bends space and time. Since both light and mass travel in space and time, they both are affected by the bends in it. You can make a model of this by rolling marbles on a flat table or on a bent table. The curves of the bent table would affect the marble, no matter what its mass is. Here is another "thought experiment." One of Einstein's postulates about gravity is that gravitation cannot be distinguished from acceleration. So imagine that you are in a spaceship that is accelerating "up" and fire a laser at the opposite wall. Because the spaceship accelerates up, it would appear to you that the laser would hit the opposite wall at a spot "down" from where you aimed it. But by Einstein's postulate, this must be true for gravity also.
This is a very short and shallow explanation of a very deep and wonderful subject. Maybe you will be interested to read a book about the theory of relativity now!

Einstein discovered that mass and energy will warp the surrounding space time. The motion of anything, regardless of its mass, moving through this region will be affected. The way we perceive how the effects of the curved space time is that objects accelerate toward each other. The force that appears to cause this acceleration is the force we call gravity.
But an object's acceleration due to gravity doesn't depend on its mass; it only depends on the curvature of the space time around it. That's why when you drop two objects, they both accelerate toward the Earth at the same rate even if they have different masses. The same thing applies to a photon or a particle of light. Its acceleration doesn't depend on its mass but only on the curvature of space time.