I'm going to take a dip into speculation, as neither Canon nor Legends really talks about any of this.
Where did Finn get any thoughts regarding the Force?
Stories, bar room chat, barracks yarns. Stormtroopers are obviously humans; conditioned humans, but humans nonetheless. They talk about landspeeders on screen in the movies; they obviously talk.
The biggest event in the universe? The toppling of the Empire by the "New Republic." And those conversations would have to include discussions about the main figures; Darth Vader, Palpatine, Luke Skywalker, Han Solo. Stories would trickle down about Vader throwing people across rooms, Palpatine predicting the future, and Skywalker beating them BOTH.
"The Force" would be something very powerful people used during the fall of the Empire, and he'd have heard about a guy named Solo being around all these "Force" people.
Where did he get the idea the Force would help them? And in what way?
I'm not a Muslim, but I have some vague ideas of what the Muslim faith might include, much of which is probably incorrect. Finn's in the same situation. The Jedi and the Sith are complex religious orders centered around superhumans with superpowers. He'd have scraps of data to work off of, but nothing solid.
Finn would expect "the Force" to help them, because he had Han Solo, and Han Solo had been involved in the toppling of the empire, where "the Force" was claimed (at least by New Republic propaganda) to have been involved in the event.
Remember, Finn is a Stormtrooper. He knows what a bunker on Endor would be like. He knows how big and powerful a legion of troopers is. He is fully indoctrinated in the concepts of the Empire as a neigh-invincible behemoth against which none can stand. And he knows it LOST, against something that claimed to have "the Force." This would awe Finn; "the Force" must be powerful indeed!
So Finn thought of the Force as just some magical "something" that made battles easier for believers. It was "something" which allowed you to blow up two Death Stars and destroy millions of Imperial Troops and trillions of credits worth of Imperial Hardware. He didn't know how it worked, but he was sure it DID work, and that Solo could do something with it.