From the comments on the original post, I gather the grammar question has been answered via editing, so this just general word usage for whereupon.
In this context, whereupon means that the action after the conjunction is happening just after—and possibly in response to (or as a result of)—the action before the conjunction.
Breaking the sentences apart [and adding some extra literal parts], we have:
Ivan laughed and said: "Well, I will go and arrange matters so that Tarras may have his share."
[ Ivan went and arranged matters. ]
[ Then, immedately after that, and because of it, ] Tarras took the brown mare with the grain to town, leaving Ivan with one old horse to work on as before and support his father, mother, and sister.
Whereupon, by the way, is a largely obsolete word in English. For whatever reason, its popularity has declined over the course of the last century; as such, you'll mostly find it in old books, or contemporary writing trying to sound old.
The following sentences without whereupon mean about the same as the quoted one:
Ivan laughed and said: "Well, I will go and arrange matters so that Tarras may have his share," at which point Tarras took the brown mare and...
Ivan laughed and said: "Well, I will go and arrange matters so that Tarras may have his share," after which Tarras took the brown mare and...
Ivan laughed and said: "Well, I will go and arrange matters so that Tarras may have his share," and so Tarras took the brown mare and...
Ivan laughed, and agreed that Tarras should have his share. Once matters had been arranged, Tarras took the brown mare and...