How do polar covalent molecules dissolve in water?
"Like dissolves like." Water is a polar molecule, so any other polar molecule is likely to dissolve in it.
Polar molecules (which may or may not consist of polar covalent bonds) do not have a symmetrical distribution of charge. One part of the molecule will be more negative (called partially negative) and another part will be more positive (called partially positive).
So the partially negative part of one polar molecule (like water) will interact with the partially positive part of another molecule (like your mystery substance). This allows polar substances to dissolve each other.
This also explains why water doesn't dissolve oils and such. They are non-polar, so there is little for the polar water molecule to be attracted to.