In this situation, I believe "that" is part of the reason the phrase isn't making sense to you.
"What a great feeling that gave me..." could be rewritten as "What a good feeling the stranger's kindness gave me..." "That" is a demonstrative pronoun for "the stranger's kindness" (or "the caller's good deed" or any similar phrasing).
"Gave" makes sense because it helps to specify what caused the good feeling. Think of it in this case as a synonym for "caused" or "contributed," not "gifted." It works with things that aren't "feelings."
- "When the teacher told me that I passed the very difficult test, that gave me a feeling of relief!"
- "When Bob said the project wasn't due for another two days, that gave me a chance to check it over again."
- "Susan had come down with a cold. That gave Alice an excuse to go visit."
It's a slightly quirky idiomatic usage that even many native speakers might not be familiar with.
What the writer means is that these "incidents" form the basis of interesting tales that can be recounted repeatedly in after-dinner conversations by a skilled raconteur (by implication, such as himself).
Because people like to be entertained in this way after a (formal) dinner, the host / organisers of the event would be likely to invite the writer. And there's usually the implication that any such "after-dinner speaker" would be likely to actually be paid to turn up (in addition to getting a free luxury meal, whereas the other diners are probably paying for the privilege of being there).
Sometimes there's no suggestion of payment / free food. I dined out on that story for years might simply mean When I went out for a meal with friends, I often used to tell that amusing anecdote, even if the speaker always scrupulously paid at least his fair share of the bill. He might just mean that his ability to entertain fellow-diners ensured such evenings were a success, and/or that people invited him out to dinner for this very reason.
Note that this idiomatic usage often includes other prepositions...
He dined off that for years
He dined out off that for years
He dined out off of that for years
He dined out on that for years
He dined out with that for years
All those (and probably more) seem fine to me, but others may feel different.
Best Answer
That sentence doesn't really make sense on its own. On Facebook, you can add certain tags to your post, and it writes those as a sentence: Feeling [x] to show an emotion. with [person/people] tags one or more people in your status. And at [someplace] tags a location. And then Facebook renders it as in your example: Feeling [x] with [person/people] at [someplace].
The word proud here is most likely related to the following definition of pride, from Merriam-Webster, and not one of the more negative ones.
What the person is proud of is not clearly stated in that sentence alone. It's most likely either something else mentioned in the status, or the person tagged.