To leave something behind is to let it remain in a location that you depart from. If, for example, you forget your keys when you go out of your house for dinner, you have left your keys behind: you left the house, they stayed in the house.
(Something) gives you away means that the thing reveals a fact about you, usually a fact that you would prefer to stay hidden. A classic example is someone who is playing poker with you, and who tries to bluff you; but their tendency to hold their breath when bluffing means that you just wait to see if they hold their breath, and if they do then you know they are bluffing: their breath-holding gave them away.
So in this case they are saying "don't bring a foreign accent with you when you go to America, so you won't let people know immediately that you are from another country."
The scope of usage for these two phrases is very broad. If someone shows up at work and does 3 stupid things in a row, you could ask them "Did you leave your common sense behind this morning?" If your boss then tells you that your co-worker is pregnant and the reason she did those stupid things is because she has morning sickness, you could say "I didn't realize you were pregnant at first, but the boss gave you away." (Any time you learn something about someone that they don't tell you directly, you could say that whatever caused you to figure that thing out about them "gave them away".)