Grammatical/Semantic basis for the phrase "what with"

What with has been a turn of phrase for a very long time in English. For instance, Erasmus Almer's book The alcaron of the barefote friers (1550) has this sentence:

And in his meditaci∣on of Christes passiō he would so beate him selfe, that what with teares and blood yt semed that ther ranne Ryuers of blood out of his body.

And in his meditation of Christ's passion, he would so beat himself that, what with tears and blood, if seemed that there ran rivers of blood out of his body.

About a century earlier, it appears in Sir Thomas Malory's Le Morte Darthur:

And soo sire Tristram armed hym and took his hors / & putt hym forth / and there sire Lamerak mette hym myghtely / and what with the myght of his owne spere / and of sire Tristram spere syr Lamoraks hors felle to the erthe

And so Sir Tristram armed himself and took his horse and put himself forth, and there Sir Lamorak met him mightily, and what with the might of his own spear and of Sir Tristram's spear, Sir Lamorak's horse fell to the earth.

It even crops up in Chaucer's Troilus and Criseyde, in book 2, lines 1436-42:

"And ȝif I more dorste prey ȝow as now,

And chargen ȝow to han so gret trauaille,

To han som of ȝoure bretheren here with ȝow,

That myghten to hire cause bet auaille,

Than wot I wel she myghte neuere faille

fforto ben holpen, what at ȝoure instaunce,

What with hire other frendes gouernaunce."

And if I may dare to request more than I do now, and charge you to have such great trouble, to have some of your brethren here with you, that you might better aid her cause, then I know well she might never fail to be helped, what at your urging, what with her other friend's oversight.

This last instance highlights in parallel phrases the origin of what with: what headed adverbial phrases formed with prepositions (for, at, with) in order to suggest (and this now quoting the Oxford English Dictionary, "what, pron., adj.1, and adv., int., conj., and n.," entry D.II.2b:

‘in consequence of, on account of, as a result of; in view of, considering (one thing and another)’.

As for the what itself? The what is indefinite but can be summed up as the following:

a. Introducing (a) each, or (b) only the first, of two or more alternative or coordinate words or phrases: (a) what..what, (b) what..and (†as, †so)) = Some..others; both..and; including..and; as well..as; partly..partly.

The Troilus and Criseyde quote also happens to be an example of this duplication:

what at ȝoure instaunce,

What with hire other frendes gouernaunce."

That can be glossed as both at your urging and with her other friend's oversight (going off of the older meaning) or on account of your urging and on account of her other friend's oversight (going off of the emerging one). By the fifteenth century, what with had become its own set phrase, and as the older usages what ... what and what for disappeared, what with became set idiom.

How did that what sneak in there? What is it doing?

How did that what sneak in there?

It is idiomatic and is common in two forms what with and what with (something)

Examples idioms The Free Dictionary

1990 Rosamund Clay Only Angels Forget She's had a difficult life, what with my father skiving off when I was three and leaving her without a penny.

What with the children being at home and my parents coming to stay, I have too much to do.

What is it doing?

It is a common phrase (at least it is in the UK) and it could be said to be used instead of "because of" Owing to; as a result of

Examples

"What with the war, and all, food is in short supply!"

"I haven’t had time do it, what with one thing and another".

what with; informal; used to talk about the reasons for a particular situation, especially a bad or difficult situation: Cambridge English Dictionary

The use of what in the expression what with is emphatic:

What with:

It is a construction in use in English since the Middle Ages: it means "in consequence of, on account of, as a result of; in view of, considering (one thing and another)". Centuries ago English people also said "What through..." and "What for..." which meant the same thing; but these expressions are now obsolete

(The Phrase Finder)