Complex sentence: Rosecrans offered the excuse that if he were to start to move against Bragg, Bragg would likely relocate his entire army to Mississippi and threaten Grant's Vicksburg Campaign even more; thus, by not attacking Bragg, he was helping Grant.
Split into simpler sentences: Rosecrans offered the excuse that if he were to start to move against Bragg, then Bragg would likely relocate his entire army to Mississippi and threaten Grant's Vicksburg Campaign even more. Thus, by not attacking Bragg, he was helping Grant.