How do Snowball and Boxer earn the title "Animal Hero, First Class"?

To answer this question, take a look at chapter 4. According to the text, Mr. Jones and a group of men return to the farm in an attempt to retake it. This leads to the Battle of the Cowshed, in which the humans and the animals fight for control. In this battle, it is the animals who win, while Mr. Jones and his men are defeated and forced to flee the farm.
Snowball is given this honor because of his military planning and strategy. Having studied the campaigns of Julius Caesar, Snowball executes a strategy in which he makes the humans believe that the animals are retreating when in fact it is an ambush: a number of horses, cows, and pigs lie in wait, ready to pounce on the humans. Snowball then personally attacks Mr. Jones, leaving him lying without his gun, in a pile of dung.
As for Boxer, he rears up on his hind legs and (unintentionally) kills a stable boy. This display of aggression terrifies the humans, prompting a number of Jones's men to "drop their sticks" and flee.
It is for these actions that Snowball and Boxer are each named "Animal Hero, First Class."