When solving game theory problems, how can you "assume" that the equilibrium strategy will lead to something happening? This is something that confuses me a bit when I see my book solve game theory problems.

When solving a model's equilibrium (like a subgame perfect nash equilibrium), the book will assume that a certain situation occurs in that equilibrium. For example, in a model with some firms setting prices and some consumers who have to make a decision to buy a product, it might assume that in equilibrium, all consumers buy.

Then it proceeds to solve for the actual equilibrium, while it always uses the assumption that they all buy. 

When having solved for the equilibrium, it checks that it indeed holds in this equilibrium that all consumers buy, by e.g. showing that no consumers have an incentive to not buy.

Why is this acceptable? Isn't the derivation of the equilibrium flawed because you made an unwarranted assumption?
This is a concept that I've seen confuses a lot of people when they first encounter it. However, it is completely perfect to assume that something will happen in equilibrium, use that fact to characterize equilibrium, and then check your assumption indeed holds. This is sometimes called guess-and-verify. Remember that an equilibrium has a very precise definition, it is a set of strategies that satisfy some conditions. If you find them, and they satisfy the conditions it doesn't really matter how you found them.

It might sound silly to just be guessing properties of an equilibrium and be hoping for the best. But actually, doing good guesses is hard; so there is merit to it. So long you check that your equilibrium satisfies the assumed (or guessed) property, then your original assumption is verified. 

The alternative would be, in principle, to solve a game by computing for each player their best response as a function of all possible actions of every other player. And then find the intersection of these best responses. This can be an extremely hard thing to do, even for simple games like a Hotelling model where N firms set locations and prices and then consumers choose from where to buy. So making informed guesses can be a great shortcut to find an equilibrium.