"Feeling obligated" to do something means that you feel like you have to do it. You're afraid that it would be rude not to do it.
People usually feel obligated to do things for social reasons, like because a friend or neighbor asked them to do it. For instance, if someone gives you a gift, you might "feel obligated" to give them a gift back in return. We usually talk about "feeling obligated" to do things that we don't want to do.
You can use this phrase in a sentence like this:
I feel obligated to help because, you know, he helped us out a couple of months ago.
I don't really want to go, but I feel kind of obligated.
You can also tell someone not to feel obligated to do something:
Don't feel obligated to come if you're too busy.