Forest has a lot of back story, but because he is "a bit slow", it's important to be able to tell which of the stories he is telling are "his" and which are stories in which he placed himself in a role. In order to do that you need to understand some popular history and what Forest would have seen on TV or heard on the radio.
Not having ability to tell what is real and what is a bit off won't make the move unenjoyable, it will still be a "feel good movie" about a guy doing some really amazing stuff.
But in the end, you take all the times that you know Forest didn't do the things he said he did, the times you think he might have done the things he said he did, and the times that you are pretty sure he's telling you what he actually did, and you compare them, all to realize it doesn't matter. The people on the bench still were interested, Forest still feels good about himself and you as the viewer still feel good about the stories.
That examination of why, is what makes Forest Gump a good movie. In my opinion. Why do you feel good, watching stories where you know some of them are totally wrong, and that the guy telling them is wrong. Ask yourself this. Would you feel the same way if Forest Gump was not "slow"?
Anyway back to the question. Yes, you need at least some understanding of American history and General culture to fully enjoy the movie. Not having that, you will miss out on a lot. But the understanding required is nothing more then the ability to go "that seems unlikely" in a few cases to get the effect, and that's a pretty low bar.
Again to get the full effect you just need enough to go "Hmm that seems off" a couple times, and "Nope, that can't be" a couple times. That hardly requires much of an education in American history or culture. What you can pick up off the TV is probably enough.