In model checking, you have a model and a specification (or property), and you check if the model meets the specification.
In SAT solving, you have a formula and you try to find a satisfying assignment to it.
Now, in model checking, you can conjunct the model and the negation of the property to give you one formula. Use a solver to solve for this formula. If it gives you a solution, it would mean the property is sometimes violated (since you conjuncted the negated property). Getting unsat
would mean your model satisfies the property/specification.