When you are writing about something, you often start with a general statement, then back it up with some specifics that are the proof that your original statement is valid or correct. These are your supporting details.
And so, if you are writing about any thing of your choosing in an essay that's where the supporting details occur/appear. So like you write about anything, then you need proof and a supporting detail more like an explanation, kinda thing.
hope this is helpful :)