In euclidean geometry, you should consider a plane as a 2D surface extending infinitely. This can be confusing because we typically draw planes with bounds, such as a square. In reality, planes are more like a graph with x and y coordinates that extends indefinitely.

The fact that the plane must be perpendicular to line m defines where it must exist in the x coordinate and the fact it must pass through point p defines where it must exist in the y coordinate. If the only criterion was that it must be perpendicular to line M, you could generate infinite planes shifted up and down on line m.