The "забы" part sounds like "sabe" in Spanish which means "he/she/it knows". But "ваю" sounds like "bye" which you know what means.
Then that is forget. Cool yet strange mnemonics.
I just love this word. In Polish, the same-looking word means 'to kill'. So if you say он забил всё it exactly means 'He killed everything'. A good way to remember it, by the way :D
bić and być are different words in Polish, too. The Russian word for "to forget" eventually comes from the "to be" verb modified with the prefix "за". The verb is быть in modern Russian.
From забыть you get the impefrective забывать which you see in this sentence. Забить and забивать are different Russian verbs that come from бить (beat, hit).
I've always thought it was spelled забить. Probably it just looked much better to me this way. :D Now I see that Russian's 'to kill' and 'to forget' differ in the stem. Thank you.
Забить is more like "to slaughter" (when used with cattle). Not a violent kind of "slaughter" but more like a technical term. It is also the verb we use when we want to say "to drive something into something by hitting it" (e.g., a nail into the wall using a hammer or a pile of a foundation into the ground)—or to stuff something up to the fullest by filling it with something.
As a slang, забить/забивать (+на + Accusative) is used to express "to skip (classes)" or generally to give up / stop caring about something you previously intended to do.
By 'differ in the stem' I meant убить and забыть and not that this и/ы thing is the only difference (I knew that Забить doesn't mean exactly 'to kill'=убить). I probably didn't express it clearly enough. Thanks for the explanation of the other word, though!