Detergents have nonpolar tail group and a polar head group. In solution, the nonpolar tail groups are excluded from water, so they form micelles
(the picture here is helpful). In these micelles, the nonpolar tails form a hydrophobic environment in the interior, while the polar head groups create a polar environment on the outside. Since the outside is polar, the micelle is able to disolve in water. However, because the inside of the micelle is nonpolar, it can capture nonpolar substances (such as dirt and grease) which are soluble in a hydrophobic environment. When these grease molecules become trapped in the micelles, they can be washed away with the water soluble micelles.
So, it really isn't a chemical reaction which solublizes the grease. Basically the detergent micelles capture grease and allow them to enter the soluble phase and be washed away.