Hows does this 12v relay works with transistor voltage drop?
This circuit is to make an alarm where the relay will never open until power is reset.

This is called a self-locking relay circuit. Basically, the transistor is only there to provide a one-time only current. This will shift the relay into the closed state, at which point the relay powers itself through it's own contacts. The transistor no longer has any incidence on the circuit since it's completely bypassed. 

The page just before your schematic explains clearly how it works:



Source: Make Electronics by Charles Platt 

To make it easy to understand: 

When the switches are activated, the transistor gets a positive supply to the Base. Which then makes the transistor sink current through. The coil activates on the relay, which then bypasses the transistor and supplies 12V to each side of the transistor.