the two-dimensional solution space
How much freedom does a second-order equation leave you? Think about throwing a ball: once you fix where it starts and how fast it leaves your hand, its whole flight is determined. Two numbers — position and velocity — pin everything down. That 'two numbers' is exactly the dimension of the solution space of a second-order equation: the family of all solutions has precisely two degrees of freedom.
More formally, the set of solutions of a homogeneous linear second-order equation y'' + p y' + q y = 0 (with p and q continuous on an interval) is a vector space, and its dimension is two. This means you can find two solutions y1 and y2 that are linearly independent, and then every single solution can be written y = c1 y1 + c2 y2 for exactly one choice of the constants c1 and c2 — no solution is missed, and no combination is wasted. Why two and not three or one? Because an initial value problem here needs two initial data, y(x0) and y'(x0), to have a unique solution. Each independent initial condition you are free to set corresponds to one dimension, and a second-order equation has exactly two such slots.
This is the structural fact that makes the constant-coefficient method work. The characteristic equation is a quadratic and so yields a 2-parameter answer to fill exactly this space. The general pattern generalises: an n-th order linear equation has an n-dimensional solution space. Knowing the dimension in advance tells you when you are done: once you have two independent solutions of a second-order equation, you have them all, and there is nothing left to find.
For y'' - y = 0, the pair e^x and e^(-x) is independent, so {e^x, e^(-x)} is a basis of the (2-dimensional) solution space. To match y(0) = 1, y'(0) = 0 you solve c1 + c2 = 1 and c1 - c2 = 0, getting c1 = c2 = 1/2, so y = cosh(x) — and these two equations always have a unique answer precisely because the space is 2-dimensional.
Two basis solutions and two initial conditions always match up uniquely — that is what dimension two means in practice.
The dimension is two only when p and q are continuous on the interval in question, so that existence and uniqueness hold. At a point where a coefficient blows up (a singular point), the clean two-dimensional picture can break down.