When. At the end of a study session, not at the start of the exam — after the reading is done and the page is closed. That timing is built into the wording: it speaks of what has already been read and already been memorised, so there must be something to hand over before it can be said.
The words. Astawdi'uka is the verb of the wadi'a, a deposit. In Arabic usage a wadi'a is a specific arrangement: property left with a trustworthy person, not sold and not given, remaining the owner's, to be handed back when he comes for it. The whole supplication runs on that single image. Faruddahu is then the ordinary verb for returning a deposit — not "remind me" and not "help me remember," but give it back, because it was left with You.
At the time of need. 'Inda hajati ilayh is exact about when. Not now, not permanently in the front of the mind, but at the hour it is wanted — in the exam hall, in the interview, at the moment of teaching it to someone. Anyone who has revised well and gone blank knows that the problem was never storage.
What a deposit assumes. You cannot withdraw what you did not deposit. The wording concedes this without saying it: ma qara'tu wa ma hafizt, what I read and what I memorised. It is a supplication for a student who has studied, and it does not pretend to be anything else.
Note on its standing. This is a widely circulated supplication among students and it carries no attribution to a specific prophetic report. Asking Allah in one's own words for the same thing is available to anyone, in any language.