Where are doxycycline's anti-inflammatory properties derived from?
Doxycyline inhibits iNOS and TNF-alpha expression.   It also influences secretion of IL-1beta, IL-8, and TNF-alpha, as well as other inflammatory cytokines.

[These are results collected from multiple systems.]

For appropriate references, see: Leite et al (2011). Anti-inflammatory properties of Doxycycline and Minocycline in experimental models: an in vivo and in vitro comparative study. Inflammopharmocology.   

As for the chemistry of how doxycycline accomplishes these things, I'm afraid I don't know the answer, but hopefully, this gives you a place to start.