Here is the idea.

This is the triangular inequality.

|x+y|<=|x|+|y|

Meaning the triangle you form out of these vectors follows the rule that the sum of two sides of a triangle is greater than the third side.

The exception (special case when we have equality) is when the vector, say "y" lies directly upon the other vectorx. In that case we do not form a triangle and have equality.

Thus, we need thatylies (coincides) withx. Meaning they have the same direction. And two vectors has the same direction iff they are non-zero scalar multiples of each other. Thus,y=cx