One way to be able to understand the style of Anton Chekov's one-act comedy The Bear is by analyzing it as what it is: A vaudeville-type farce that is written with the purpose of providing the audience a quick and surprising piece of theatrical entertainment.
This being said, it is expected from a dramatist of this kind to create characters that are not too prone to be analyzed in-depth. Any depth of character, or any depth in the plot itself would compromise the style and purpose of this kind of theatrical work.
Therefore, what Chekov does is to introduce two characters who have a problem in common: The death of Mrs. Popov's husband. However, we know little about their past, present, or future. What we do know is that the problem at hand is that Smirnof wants his money and Popov does not have it. The farcical part of the play comes when we realize that Smirnof dislikes women and talks bad about them whereas Popov is over-dramatic and is reacting in an exaggerated manner towards her husband's death.
These odd behaviors and their current problem is all that the audience has to determine whether Popov and Smirnof are a) crazy, b) silly, c) eccentric, or d) all of the above. Since it seems that they are, in fact, quite eccentric and silly it is no surprise that the ending will turn out to be as weird as the characters that lead it. Smirnof ends up proposing to Popov even after the two had prepared to duel to their deaths, and- like in many short and funny movies and stories- all the insanity seems to magically end with a kiss.