Mepron/zithromax caused a significant bartonella herx for me. This is likely due to bartonella's antibiotic resistance mechanism being largely dependent on efflux pumps (see
https://pubmed.ncbi.nlm.nih.gov/27667026/). Since atovaquone inhibits energy production in bartonella, this removes the power source for the efflux pumps, so bartonella can no longer eject the azithromycin.
The synergistic effect in babesia is a little less direct. Babesia contains an apicoplast (endosymbiont prokaryote) which is susceptible to antibiotics. If you can kill the apicoplast with antibiotics, the babesia parasite will eventually die. The apicoplast is inside the larger parasite, but you've got to get enough antibiotic in there to affect it, and it seems that hitting babesia with atovaquone can help with this.