Visibly means 'in a way you can actually see'. It is nearly always used about a person's feelings showing on the outside: visibly angry, visibly upset, visibly nervous, visibly relieved. The point is that the feeling was not hidden — anyone in the room could tell just by looking.

So it does two jobs at once. It says how the person felt, and it says that this was on display. 'He was visibly shaking' is stronger than 'he was shaking', because it puts a witness in the picture: I could see it, and so could you.

It sits most often in front of an adjective (visibly tired) or a verb like flinched, relaxed, stiffened, brightened. It is a little more formal than everyday chatting — you meet it a lot in news reports and novels, where the writer wants to report only what could be observed rather than claim to know what was inside someone's head. A reporter cannot say 'the minister was furious', but can safely say 'the minister was visibly furious'.

It can also mean simply 'noticeably, by an amount you can see', about a change: 'the queue was visibly shorter by noon'.