In order for the double-stranded DNA to separate and become two single strands, the forces holding the two strands together have to be overcome. The two strands are held together by hydrogen bonds, which are relatively weak interactions, but there are many of them.
Just like transitioning from a solid to a liquid requires the addition of enough energy to break the interactions holding the molecules together, melting DNA (transitioning from double to single strands) requires the addition of enough energy to overcome the forces holding the strands together.