The word originates from the habit of cat owners who try to find homes for unwanted kittens by going door to door to see if anyone wants to adopt. In the metaphor, the second woman is like the kitten that is being brought to, or "dropped" on someone's doorstep.
There are numerous variations on this theme. For example, the couple may share the "kitten". However, when the person being dropped is male, it's generally called a "puppy drop".