When Paine uses the word “try” in this context in “The Crisis,” he means something like “test.” In other words, he is saying that the times they were living in were the times that tested men’s souls. He is saying that their situation was going to test men (and presumably women as well, though people did not think they were important in those days) and see what they were made of.
We can see this meaning by looking at the next sentence at the beginning of this first pamphlet of Paine’s. He goes on to say that
The summer soldier and the sunshine patriot will, in this crisis, shrink from the service of their country; but he that stands by it now, deserves the love and thanks of man and woman.
Paine is saying that some people will not stand up for their country and their freedom. These are the people who, in today’s language, were “jumping on the bandwagon” of freedom when times were good. They are “fair weather friends” of their country.
Paine is saying that these times are going to test people out. The times are going to see who is really a true patriot and who is not. The former deserve our thanks and love while the latter deserve to be scorned. By saying “these are the times that try men’s souls,” Paine is saying that these are the circumstances where we find out who fits in each of these two categories.