How did the moving of the capital from Rome to Constantinople weaken the western part of the empire?

Answer 1

Just to follow-up on the first answer, the lack of financial resources was compounded by the Vandals taking over N. Africa which was their only source of funds after Constantinople split off. Without funds, a standing army could not be maintained in the field and the rest is history, they were soon over-run by the barbarians.

Answer 2

The capital did not move. Western Roman Empire kept its capital at Rome for quite some time.

It's to begin with that half of the empire was attributed to the Eastern Empire, so its resources poured into Constantinople, not Rome. That and the fact that with the state of communications at the time, Rome wasn't able to effectively maintain control of its Northern Gaul and Germanic possessions, not having the money to effectively garrison the limes anymore.