an insanely common verb) There is another one with a different shade, but the same meaning: выходить (lit 'come out').
1. получаться/получиться. To be a result, an outcome of something. Turn out. You'll find different translations but always it has a shade of 'result'.
"Ручки, ножки, огуречек - получился человечек" (also...вот и вышел человечек)
A children rhyme about drawing stick figures: "arms, legs, a cucumber - and we have a man'.
Я приготовил свинину, а гости оказались мусульманами. Получилось неловко.
"I cooked pork, but it turned our the guests were Muslims. It was a bit awkward"
We use it freely and learn in a very early age.
2. (у меня/тебя/неё/него/...) получилось - you accomplished/succeeded in somthing, achieved the outcome
needed. не получилось = you failed )
Imperfective получаться - "each time you try" "mostly" - OR if the task is somewhat complex, and everythin goes right at intermediate stages and it seems you will succeed. Eg. while threading a needle:
"У меня не получается!" = I can't! (said while trying or after).
"У меня получается... погоди-ка!"
"Получилось!"
3. У меня хорошо/плохо получается.
Not the Most basic way to say 'good at something'. A shade here: when he_ tries_, when he is doing this.
An observation.The context must explain this: Either it is just an observation. Or the task is rare enough, so you don't want to think/speak of a person as having a 'skill" in this strange thing:) Or he tried just a few times?
"он хорошо..."=he is good at... is an immanent quality, like having a skill.
When said about yourself, it is more humble. 'I see that when I try, the outcome is good'. Я хорошо.. is more "bragging".
When said about other person, it still can be preferable. What if you don't want to sound insisting that somebody is good at something?