The reason Washington would do this was to raise the morale of his soldiers.
The first essay in The Crisis was dated December 23, 1776. This was one of the darkest times in the Revolutionary War. Washington and his army had been pushed out of New York and through New Jersey. They had ended up in Pennsylvania, very discouraged. Even the Continental Congress, which was the US government at the time, had been forced to run away from the British army.
In this context, it seemed that the Revolution might well be over. Washington would have needed a way to encourage the men to continue fighting. One way to do that would have been to read Paine's work to them. This essay is a very powerful statement of why it is important not to give up the fight. It does things like encouraging the men to keep fighting because their victory over tyranny will be glorious.
Tyranny, like hell, is not easily conquered; yet we have this consolation with us, that the harder the conflict, the more glorious the triumph.
Because of the fact that it was inspirational, Washington would have wanted his soldiers to hear it.