How does Time Machine work for a photographer
Could someone explain, in detail, how the back-up process works? 


Yes. Nobody better than the late Pondini, however: How Time Machine Works its Magic. Essentially, the first time you backup a machine, a full clone of the internal drive (with the exception of a few unneeded directories) occurs. Thereafter, every hour an incremental backup occurs, where all changes to the drive are backed up. For everything that hasn't changed, hard links are created to the full version . After 24 hours, hourly backups are pruned and consolidated into a daily backup. The same process extends to daily backups after a week (all daily backups from that week are consolidated to one week).


  I'd like to be able to figure out how large of a back up drive I should get.


It is recommended to purchase a drive at least twice the size of the drive (or cumulative size of drives) you are backing up. 


  Is it backing up/creating a new image file after every tweak of every knob I make in LightRoom?


No, not Time Machine at least. Only hourly. Every hour, time machine reads changes from the file system events store and backs up any folder in which file changes within it have occurred. 


  is this how lightroom works? It comes with a "history" feature and I
  wonder how it is I can revert a photo on lightroom if it was to not
  create a copy over and over while I'm editing.


I can't speak to exactly how lightroom works, but in doing so it's taking advantage of the same features Time Machine is taking advantage of: an HFS+ (journaled) file system in which changes are routinely catalogued by OS X itself. This allows programs to retain previous versions of files in their cache. Even TextEdit, for example, allows one to revert to previous versions without Time machine even enabled.