"They" give the president nothing. There is no "they."

The president nominates a chairman from among the seven sitting governors, under the Banking Act of 1935; this is subject to Senate confirmation. No different, really, than any other presidential appointment. 

That also keeps the chairman and the board of governors somewhat more insulated from the political fray. Can you imagine the chaos of monetary policy under straight-line democracy, changing willy-nilly as the political winds shift? I consider myself better informed than, say, most other schlubs on the streets, but I'll admit I'm not expert enough to know who would have the best view on the federal funds overnight benchmark rate and when is the right time to adjust it.