In the case of melting points (MP), MW may be used as a good guesstimate when you don't know all the variables. Intermolecular forces determine MPs and BPs. In the case of MPs, you go from solid to liquid and molecules are much much closer to each other (in comparison with the gaseous state). That's why when everything else is about the same MWs are usually good MP predictors (when everything else is equal the more compact your substance the higher the MP of that substance). You can't always use MWs to guesstimate BPs because there are too many exceptions and molecules are not very close.
because size increases as you go down a column (so does MW)
P.S. Other factors also play important roles (e.g. branching, charges, hydrogen bonding, polarity,...)