Learning a second language opens doors for you, both here and overseas.
Imagine being in a foreign city. You see people walking the sidewalks; you hear the "vroom" of the cars, motorcycles, and mopeds on the streets. Further down the sidewalk, you come across something that smells real good, your stomach churns. You step into one of the restaurants.
You sit down and the server hands you the menu. When it's time to order, you find yourself doing something like this, "what's this entry like?" Immediately, the server goes 100 mph in the native language. You're getting hungry, but you're not ready to order something that might take your appetite away.
If you speak the local language, you'd be able to read the store signs and what they sell. Once you step into a restaurant, you could read and understand the menu; then place your order. When the natives see you speaking in their tongue, they remove you from the stereotype they have of people from your home country. They end up liking you more and go out of their way to help you out.
Getting into conversations with people that don't speak your language is a learning experience. The person you're talking to will tell you things that you won't get out of a guidebook. You might find a place where you could knock your gift shopping, for the family, out in one afternoon; and have money to spare. You might find a hole in the wall place where no tourist ventures; one that's safe and where decent locals like to hang out at.
A good language to learn is one that's spoken in multiple countries. Spanish, Portuguese, and Arabic are examples. You remove most language barriers for multiple countries just by learning one language. There are regional differences and accents, but you'd be able to overcome these in a few hours.
Language barriers have the potential of ruining people's stay or vacation overseas. If you get lost, and nobody could understand you, you open the door to someone taking advantage of you.
Finally, the number of people speaking a foreign language in the U.S. is increasing. You may end up in a restaurant, deli, or establishment where the employees don't speak English. Knowing the employee's language could make the difference between having a decent meal; or you complaining about a bad experience to friends.