To be honest, this is something that not even most astronomers fully comprehend, and I don;t pretend to understand it completely myself. But I'll try to muddle through an explanation as best as I can.
Imagine you and I were to start in opposite directions from a single starting point and you had a gun capable of firing a bullet that would travel at a constant speed until it hit something. And let's say we both starting traveling at the same time and could instantly travel at a speed just 1/2 the speed of the bullet the gun you had would fire. Let's also say that we both traveled for one hour before you fired the gun in my direction and then dropped the gun at that point but you continued to travel at the same rate. But since we are both traveling away from the point where the gun was fired, by the time the bullet caught up with me, we would both be a much greater distance from each other than when you fired the gun.
What you have to keep in mind is that the speed of light is constant, but it is constant from it's place of origin. While light may be traveling at you at 186,000 mps, if you are traveling at away from the source of the light, it will take longer to get to you than if you were standing still. And the universe is not only still expanding, but evidence indicates the expansion is actually speeding up.
I know this is probably not the best explanation, but I hope it helps.