Why does strength of bond matter if melting only affects the the intermolecular of the molecules
Depending whether you have a molecular, metallic or ionic compound, the independently moving particles in the liquid state are molecules, atoms and ions, respectively.
So for a molecular solid, you have to break the intermolecular forces to turn it into a liquid (or make them non-persistent so that the interaction partners can change over time).
For a metallic solid, you have to break the metal bonds intermittently so that the atoms can move in the liquid.
For an ionic solid, you have to break the ionic interactions intermittently so that the ions can move in the liquid.
Finally, for a network covalent solid, you would have to break covalent bonds to liquify it. For this reason, there is no liquid state of diamonds.

So why is it that metal groups decrease in melting points going down [...]

According to Jim Clark:

The strength of a metallic bond depends on three things:

The number of electrons that become delocalized from the metal
The charge of the cation (metal).
The size of the cation.