In The Frogs, the rivalry between Aeschylus and Euripides is especially comedic because the grandeur of the former and the emotional intensity of the latter are reduced to absurd tropes, with each playwright savagely parodying and satirizing the work of the other.
Euripides begins by coining strange, incongruous words such as "unperiphrastic" and "bombastiloquent"* to describe Aeschylus's grand style. Aeschylus retaliates by calling his rival a "chattery-babble-collector" and a "rag-and-patches-stitcher." Each then undercuts the other's recitation of his finest passages with ludicrous stock phrases.
The comedy here lies first in seeing the two distinguished tragedians behaving so childishly, subverting their own images as they mock one another's work. Then there is the fact that there is an element of truth in their critiques, which is all the funnier coming from a fellow tragedian. Finally, there is the characteristic Aristophanic technique of bathos, in which speeches are built up with great eloquence only to be brought crashing down by the absurd intervention of the other dramatist.
*References are to the Harvard Classics edition, which is available on Project Gutenberg and linked below.