Why do bulls-eye rashes look like they do?
The spreading of the rash is caused by dissemination (migration) of the Lyme spirochetes (Borrelia) through the skin out from the initial site of infection. 

The spirochetes trigger a immune reaction that involves macrophages entering the skin and releasing pro-inflammatory factors. 

The clear areas (when present) apparently represent areas where macrophages have been cleared from the skin and thus are no longer triggering a inflammatory response.

The different patterns associated with the rashes can be explained by differences in how fast macrophages are cleared from the skin.

Reference:
Vig, D. K., &amp; Wolgemuth, C. W. (2014). Spatiotemporal evolution of erythema migrans, the hallmark rash of Lyme disease. Biophysical journal, 106(3), 763-768.