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.
Context and tone of voice are key here.
A: blah blah blah and I am so glad to be done with that.
B: Yeah, I hear you. (Sympathetic)
A: And remember to blah, and then blah, and it's really important to blah.
B: I hear you! (Stop nagging me; impatient)
In my experience (US) the former is more common but the latter sometimes occurs.
"I hear you" is equivalent to "I see" or "I see what you mean". It means the speaker believes he gets your meaning. It does not imply impatience.
As an aside, people tend to prefer one of their senses for use in metaphorical phrases like "I hear you" or "I see". Some people "hear", others "see". Occasionally someone will even say "I feel you". Hypothetically, a sentient being with different sense priority could be expected to say "I smell what you're getting at" or "I taste your point"!