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I really have no problems understanding what the phrase means, so please do not answer with a description of the phrase.
I was looking for the grammar behind this: Why, instead of any other preps, we have "at" used in this expression? My research didn't reach any results. I would write the questions relevant to give direction to what answers I seek.
At it is idomatic, as I suspect you know. The OED defines at it under at as:
16b at it: hard at work, fighting, etc.; busy.
I point this out because I think the etymology of the idiom can be gleaned from the nearest definition of at:
16a: With actions in or with which one is engaged: as at dinner, at work, at play.
In other words, at means to be engaged with, or performing actions related to, a thing. Using this definition, I think the idiom can be constructed with it either being something known from context, or a generic reference (meaning, tautologically, "whatever it is you are currently doing").
So, "while you're at it" can mean "while you are engaged in [something apparent from context]".
I'm going to get a loaf of bread from the store.
While you're at it, get a gallon of milk.
The second sentence meaning, "while you're at (or, "engaged in") the task you mentioned", [do this related thing].
From this somewhat literal sense, we can easily jump to a more figurative sense meaning "to be engaged in that with which one is engaged", i.e., "busy", generically.
Edit:
To answer your other question, the only other preposition that I know of that might be used in at's place is about. Again, the OED, 11a of about:
Occupied with, attending to; dealing with; interfering or meddling in; attempting...
including:
while one is about it: while one is doing something already undertaken, so as to save time or effort or as a useful addition.
This use of "at" is not so much a matter of grammar, you find it in dictionaries. Longman DCE has it in at, no.15 (to be at it) and no.16 (to be at it again, disapproving).
http://www.ldoceonline.com/dictionary/at
Just a word meant as a kind of explanation. Today there is a discussion about the history of the continuous forms (to be doing). Traditionally "doing" is seen as a participle, though most neighbouring languages (Dutsch, German, Scandinavian) have a construction with prep + gerund. And in older formulas as We went a-hunting/we were a-hunting "a-" can be seen as a remainder of a prep (preposition). The prep might have been "at/in/on".
In such expressions with "at+activity" such as to be at (doing) it (again) you might see older usage of to be at + gerund. In continuous forms today no prep is used and the feeling that there was once a prep has been totally lost as it seems.
I know that this view is not generally accepted and in the history of the continuous forms constructions with participles were also used at an early stage, parallel to gerund constructions and both constructions intermingled.
All the same, perhaps this explanation diverging a bit from topic might help you to understand this special use of "at" - that astonishes you so much - a bit better.
Think of it as a contraction of:
while you are [working] at it ...
or
while you are [looking] at it ...
