Why is a methyl group a mildly activating group but a methoxy group is
strongly activating?

The oxygen on the methoxy group has electrons that are not involved in
bonding. The benzene ring can get some share of those electrons. A
methyl group has no unbonded electrons. It is only electron donating
because the electrons that make up the C-H bonds are slightly
available to the aromatic ring through hyperconjugation.