This phrase means that two things look good together. It's a similar idea to "matching". You could also say in this situation:

This color really matches your skin tone.

But the word "match" can also mean that two things look the same as each other. If your shirt and shoes "match", it might mean that they're the same color. But if you say that they "go well" with each other, it just means that they look good together:

Those shoes go well with that shirt.

Another variation of "go well" — "go well in ___" — can be used to talk about decorating a room:

Do you think this will go well in the study?