what does the preposition "up" mean in "He got beaten up by other kids in school"?

Solution 1:

Up as a particle in verbal idioms often has an energetic or intensive sense, roughly equivalent to something on the order of "thoroughly" or "purposefully". Some examples:

act up, beat up, break up, clean up, dress up, eat up, 'fess up, fill up, gang up, head up, jazz up, mess up, mix up, open up, pair up, rub up, sober up, screw up, shut up, team up, use up, wrap up.

Solution 2:

StoneyB's answer is correct, but leaves out an important point, which is the specific effect of the intensification provided by up in this case. It implies thoroughness and purposefulness, but to what effect?

What does it mean if a student is beaten at school (leaving aside the sense noted in a comment by Peter Shor, of being defeated in a competition)? In a school where corporal punishment is used, it most likely means that the student was struck by a teacher or administrator. Serious injury is less likely to have resulted. If the beating is given by another student, we would be left without a sense of the outcome: A beat B repeatedly might be followed by and then B pinned A to the floor and made him say "uncle."

On the other hand, if a student is beaten up, it is much more likely to have been done by other students or outside ruffians, and the student will have been visibly injured or otherwise unambiguously defeated by the assailant. The student would not be expected to have the capacity to overcome the assailant. Beaten up implies a more conclusive outcome.

Solution 3:

The particle "up" in "beat up" is a telicity marker.

Telicity is a property of the verb that can be simply rendered as completed-ness of the action expressed by it.

The book Particle verbs in English by Nicole Dehé quotes (p. 61) L.J.Brinton (1985):

[Particles] may add the concept of a goal or an endpoint to durative situations which otherwise have no necessary terminus. That is, the particles may affect the intrinsic temporal situation and hence alter its aktionsart from atelic to telic.

(emphasis mine)

Thus, one can say She ate the meal for hours (atelic), but not *she ate up the meal for hours (here the telic marker "up" conflicts with the imperfective timespan "for hours").

Solution 4:

Aside from the already present answers (and expanding on StoneyB's answer) this could originate from the fact that English is a Germanic language. There are many similarities between English and German and one of them is what in German is known as a "separable verb".

First you need a "separable prefix". An example is "an", a preposition meaning "at" or "on". In other contexts it can mean "on", such as describing a light.
You then take a verb (such as "machen", or "to do").
Merging the two, you get "anmachen" meaning "to turn on"

Similar to how there isn't a German verb for "turn on (a light)" but instead there is a modified version of an existing verb, there isn't a single English verb for "to give up". You can think of this as being formed from "up" and the verb "to give", both of which have meanings completely irrelevant to the verb they form together.

I think you can say that "to beat up" is another one of these verbs. "To beat" means "to defeat" or "to strike" but with by adding this particle you can change the meaning to something closer (but not exactly) to "to bully" but not nearly as violent as "to beat".

The only way to know what these verbs with particles mean is by remembering them as you would remember any other word.