Imagine a world with $6$ people.
Assume a graph, where each person is a vertex and there's an edge between two vertices, if the two persons representing the vertices like each other.
So, the question represents a scene where everyone loves everyone else, and there's no conflicts of gender whatsoever (because the graph is complete, and not bipartite).
We, the gods, as a wedding planner, have a job to unite those who like each other. If we do our job perfectly, there's no lonely person in the end, and every person is married to exactly one person. We call it a "perfect matching".
If everyone likes everyone else, then it's very easy for us.
We pick first candidate to marry, there's $6$ choices for that.
Now, we pick second candidate to marry the first one we picked, there's $5$ choices for that.
So, total possible matches are $6 \times 5 = 30$
But, who we picked first, and who we picked second doesn't matter, because in the end, they're going to be the same couple.
So, total possible matches will be $ = \frac{30}{2} = 15$