I'm assuming you mean in three dimensions (where the plane definition is ax + by + cz + d = 0 for some real numbers a,b,c,d).
The reason why it can't be a point is because the intersection of two planes (in 3D) can only share a common line, be the same plane, or be parallel (and have no intersection at all).
If you want the mathematical reason why then set up a system of equations using a matrix formulation of Ax = b where A is the [a b c] portion of the plane and b is the d portion of the plane equation and reduce the matrix.
You'll find that you have a 2 x 3 matrix which means that you will always at best have one free parameter. Because of this, you will always have (again at best) a line since a line is defined to have one free parameter.
The mathematical terms that describe this are known as rank and dimension (look them up for a clearer understanding) and the idea is that if you want a unique answer to a linear system then it means that Ax = b has a unique solution x if and only if A is square matrix (so 3x3 and not 2x3) and is also invertible (non-zero determinant).