How is humor brought out in the play The Bear by Anton Checkov?

The two primary sources of humor in Anton Checkov’s one-act play The Bear are situational irony and hyperbolic wordplay.
Situational irony is when an audience has the reasonable expectation that something will occur, but their expectations are upended. Audiences often find situational irony humorous because it is surprising. In Chekhov's comedy, the recently widowed Elena Popova is accosted in her home by Grigory Smirnov, a landowner and a creditor to her late husband. When Smirnov demands 1200 rubles from Popova, she responds angrily and orders him to leave. When Smirnov refuses, tension escalates to the point of a farcical duel. By the end of the play, Smirnov realizes that he has fallen in love with Popova and kisses her. Popova kisses him back. This is situationally ironic because these characters have been raging against one another for almost the entire length of the play but suddenly fall in love at the end.
Checkov is also a master of using hyperbolic language and wordplay to comedic effect. In The Bear, both Popova and Smirnov complain about one another in overblown terms humorous to the audience. For example, examine this collection of Smirnov’s lines.

“I’d rather sit on a barrel of gunpowder than talk to a woman.”
“Hang me on that nail feet upwards, if you like, but have you ever met a woman who can love anybody except a lapdog?”
“All women, great or little, are insincere, crooked, backbiters, envious, liars to the marrow of their bones, vain, trivial, merciless, unreasonable.”

Smirnov does not intend these lines to be literal but rather to express his frustration with Popova. Comedic lines such as these pepper the dialogue of The Bear and are a major source of humor.
I hope this helps!