Can you explain the resistors in this PIR circuit?
R1 protects led 1 from dying. 

R2 protects q1 from dying. It also sets the base current that is multiplied by the gain of the transistor to allow that much current through the C-E junction.

Diodes like led 1 and the base of a transistor must be current limited so they don't go into a overload condition and burn out. 

The delay is completely internal to the PIR module.