This cloud of counterions lowers the effective charge density and relieves the repulsion between the strands. As for the effect of salt on hydrogen bonds, you have to realize that the hydrogen bonds formed between bases in duplex DNA contribute little to the stability of the duplex. Entropy and enthalpy, of course. Enthalpy in the form of favorable, stabilizing interactions between the pi electrons of the aromatic rings of bases as they stack on top of one another. Correct base pairing is nice, but doesn't add much.
Incorrect base pairing, on the other hand, takes a lot away.