At the end of a certain teacher's video lessons, the teacher always says 以上で今日の勉強を終わりにします, and while I understand the sentence as a whole I don't grasp how exactly 以上で works in it. Could someone explain and provide an extremely literal translation of the whole sentence?
An extremely literal translation of 以上 would be "the above". When used as a standalone word (i.e. not preceded by a quantity), it refers to what has been said up to the point it is uttered. In the case of (horizontal) writing, "above" can be understood quite literally.
Normally, what is referred to has some length, like an argument as opposed to a single statement. If 以上 follows a short statement, it has a similar effect to reading out "period" in English. You are basically saying you got nothing else to say.
In your case, it refers to the whole of what is taught or discussed during the course of the class. An extremely literal translation of the whole sentence would be something like:
With the above, (I/we) take today's studies as finished.