When David had finished offering up the burnt offering and the fellowship offerings

This is a metonym for David directing the priests, who performed the actual sacrifices. (See: rc://en/ta/man/jit/figs-metonymy)

he blessed the people in the name of Yahweh

To bless "in the name of Yahweh" means to bless with Yahweh's power and authority or as his representative. (See: rc://en/ta/man/jit/figs-metonymy)