I found this article ([free PDF on ResearchGate) from 1989 in Behavioral Ecology and Sociobiology entitled "Fighting, assessment, and frequency alteration in Blanchard's cricket frog". The author analyzed the sound wave frequency of the frogs' dominant call to see if it negatively correlated with fighting. In other words, were the frogs with deeper calls better fighters, and did other frogs use that to decide whether or not to fight?
Larger frogs have deeper calls, and smaller frogs tended to retreat from broadcasts of a synthetic low frequency call, which indicated a large male. In contrast, frogs attacked a high-pitched call, which indicated their "opponent" was smaller. He also found that small frogs modulated their call to appear bigger to their opponent – they bluff.
Fighting (which was apparently only between males) was initiated when one frog was calling and another nearby kept calling after a series of calls from the first that were increasingly aggressive. The first frog would search for his opponent, and
When contact was made, a male moved onto the dorsum of his opponent and attempted to grasp the lower male with his forelimbs. Contact rarely lasted for more than a few seconds and was terminated by the lower male jumping away. Such wrestling bouts were followed by another period of aggressive calling, continued wrestling, [and terminated] by either retreat or satellite behavior by one of the males. Wrestling contests were generally short, lasting from 1–18 s (N=15, $\bar x$=7.2, SE=1.42), and they were composed of from 1 to 6 bouts (N=9, $\bar x$=2.7, SE=0.58).
So, it looks like these particular frogs fight by wrestling until one of them either leaves or adopts submissive "satellite behavior". I don't know if this is representative of all frog species, but it sounds reasonable.