Quotient spaces until recently gave me a great deal of trouble. I had no intuition for them, and no image to latch on to.
Until I started taking algebraic topology.
I learned that the quotient under an equivalence relation is like gluing 2 spaces together, where the points you stick together are identified with each other.
This excellent image illustrates the idea for the attaching map of two topological spaces. The requirement that $$f$$ be a homeomorphism is also clearer from the picture.