When Paine says this, he means that the time he was in (the second winter of the Revolutionary War) was a very difficult time. In this context, to "try" means to "test." Paine is saying that these difficult times will test people's souls, determining who is a true patriot and who is not. Paine says that this test will determine which people only support the cause when it seems to be winning (these are the "summer soldier and the sunshine patriot") and which truly have the moral fiber to back the revolution even when things are going badly.
Paine saw that the revolution was in a crisis and he was trying to encourage more people to stand up and support it so it would not collapse.