I'm going to answer this in terms of my own personal experiences:
1) You get to see how the rest of the world lives. Going to a foreign country is an eye-opener to how people interact with one another, how our country is run and the ways it can improve itself (usually, wine is cheaper than water; 6-week holiday for workers; beaches are topless; siestas; eco-friendly)
2) You can be whoever you want and no one cares. You’re the foreigner now. You already look out of place, so why not take it to the extreme? When I went backpacking, I turned into a full-fledged hippie- all bandannas, moccasins, and tattered clothing- I even forgot to care about routine showering (it’s all a part of the look). Give yourself to the experience, and you could come back a completely different person, in a good way.
3 America needs a national train system. The countries in Europe are packed as closely together as our states, and travelers can hop from place to place in no time. It requires basically no planning ahead, you just grab a ticket and hop on a train wherever you want, whenever you want. No customs, no huge fees, no giant suitcases. Dreamy.
4) It shows you how you can improve yourself. People in Europe speak, like, five languages each. Talking to foreigners highlights certain things we might want to improve in ourselves. But remember, you can teach them as much as they can teach you.
5) Makes you appreciate your own country. Exploring foreign territories is exciting and wonderful but inevitably there will be that tug of homesickness. By the time you return to the states, no matter how great your overseas experience was, you will be hugging your Starbucks venti and standing inside free public bathrooms just because you can.