While excellent answers have already been given, I think you might like some more concrete steps. Say I have two solutions, $y_1$ and $y_2$. If the equation is represented by some linear differential operator $D$ (it could be $D=\frac{d^2}{dx^2}+x$, $D=x^2\frac{d^2}{dx^2}-\frac{d}{dx}$ or any number of things), then
$D(y_1)=0$ $D(y_2)=0$
these are zero precisely because $y_1$ and $y_2$ are both solutions to the homogeneous ODE by assumption. So clearly
$D(y_1)+D(y_2)=0$
and if the operator is linear (which it is here)
$D(y_1+y_2)=0$
So if you have two solutions, by linearity their sum is a solution as well. At its core this is just a fancy application of $0+0=0$. A very similar statement can be made about scalar multiples. Why two independent solutions are needed to find the rest (i.e, they form a basis) has been answered excellently by others.