The short version: It's all explained in The Shattering, a novel released recently in the leadup to Cataclysms release. WoWinsider did those of us not inclined to read the Warcraft novels a favor by providing by-character synopses of the novel. Here's the rundown on Thralls story and why he abdicated, and Here's the meat on Garrosh's early days as Thralls chosen successor, and the somewhat rocky first few months as Warchief.
The Really Short Version: Thrall left Azeroth on a pilgrimage to Draenor when the Cataclysm was approaching to investigate stirrings among the elementals - Garrosh, as his second, was named 'Acting Warchief' in Thralls absence. In the aftermath of Deathwings return, Thrall has applied his efforts primarily to working with the Earthen Ring and the Council of Tirsifal to eliminate the threats of the Twilights Hammer, Deathwing, and hordes of extremely angry Elementals after the Cataclysm. Meanwhile, in the wake of the Razing of Ogrimmnar and the elemental invasion, Garrosh, already a war hero from the campaign in Northrend, has become even more popular with the Orcs of the Horde, as well as the Tauren, now led by Baine Bloodhoof, cementing his power.