the solution space of a system
Every solution of a homogeneous linear system is just one motion of the same machine, started from a different state. Collect all of those motions into one set and you get something with a beautiful hidden structure: not a jumble, but a vector space, with a definite dimension you can count.
Consider the homogeneous system x' = A(t) x with n unknowns, on an interval where A is continuous. Its set of solutions is closed under addition and scaling: add two solutions, get a solution; multiply one by a constant, get a solution; the zero function is in it. Those are exactly the axioms of a vector space — but a space of functions, not of points. The dimension theorem then says this space is exactly n-dimensional. The reason is a clean correspondence: by existence-uniqueness, each solution is determined by its value x(t0) at one instant, and that value can be any vector in n-dimensional space — so solutions are in one-to-one matched with R^n, which has dimension n. A fundamental set is precisely a basis for this space.
Knowing the dimension is n is what guarantees you have found everything. Once you produce n independent solutions you can stop — no surprise extra solution is hiding, because the space is no bigger than n. This is the structural backbone behind every solution recipe for linear systems, and it is why 'the general solution' is a finite combination rather than something open-ended.
A 3-by-3 homogeneous system has a 3-dimensional solution space. Find three independent solutions x_1, x_2, x_3 and you are done: every solution is c1 x_1 + c2 x_2 + c3 x_3, and the three constants are fixed by the three components of x(t0).
Dimension n means n independent solutions span everything — and counting to n tells you when to stop.
The dimension equals n only for the homogeneous system. The solution set of a nonhomogeneous system x' = A x + g is not a vector space (it lacks the zero solution) — it is an affine shift of the homogeneous space by one particular solution.