Can anyone tell me how working through the 12 steps removes the desire to drink?
Yes.
The 12 step process, which never really ends, removes the need for alcohol & drugs in a person's life.
There is no guarantee the 12 step process will work for you.
But if you don't try the 12 step process there is a guarantee it won't work for you.
(Choices, choices, choices... that's all we seem to ever face... choices. 
)