You use "I feel like ___" to express your opinions. When you state your opinions this way, it sounds more personal than when you say "I think ___":
I feel like kids these days don't have the same sense of respect that they used to when I was growing up.
You can also use "I feel like ___" when you're not quite sure of something. For example, if you lost your credit card:
I feel like I might have left it at the bar I went to the other night.