My culture is the culture of England - my travelling will not improve the culture of England. (Though, God knows, it could do with some improvement.)"travelling will improve your culture"
Regarding the truth or not of this idiom: in my context it is supposed to be true (hopefully) being an education program/exchange organised by an university XDTo me "Travel opens your mind" is challengeable. Has it been proven? "Travel broadens your horizons" is a truism in the sense that the more you travel the more horizons you see, and "broadens your horizons" is another way of saying "opens your mind" so even if travel doesn't actually improve your ability to absorb culture then no one can argue that it doesn't literally broaden your horizons.
Hope that makes sense.