Is "I hear you" impatient or sympathetic?

I have several times come across the phrase "I hear you" (spoken in response to, e.g., someone relating an experience of theirs), but I haven't been able to figure out whether it signifies a polite impatience or whether it implies understanding (or some of each). How should I use this phrase, and how can I interpret it?

All that depends on the context where "I hear you" is used. It is as good as "I see" kind of a remark.

If "I hear you" is used in response before you complete a sentence then may be you are tautologic with respect to the other person.

If it is used after you complete a sentence then it is very much a kind acknowledgement.

In the UK it occurs as 'I hear what you say', which means 'You're talking a load of garbage, but I'm too polite to say so.'