The phrase "as well as" means "in addition to", e.g. in the sentence "I'd like a banana as well as an apple".
I did some Googling and found it's the modern spelling of the archaic "aswell"[sic] (no space present) but I cannot find any more than that. Is it a coincidence that the "well" in "aswell" is the same "well" used for well (where you get water from), or well (to be in a good state)? I find it too coincidental that "aswell" contains two modern English words "as" and "well". How did "aswell" and now the modern spelling "as well" come to mean "in addition to" as neither of these meanings lend themselves to morphing into that as far as I can tell.
I suppose more things is better so maybe the second meaning of well could lend itself to the etymolopgy of this phrase.