solvent just means a medium in which something dissolves. in this case solvent dissolves the gunky grease. when people say solvent they usually mean "organic solvent".... a nonpolar liquid that will dissolve other nonpolar stuff.

actually degreaser has a solvent too... water. but degreaser works by essentially making greasy things more soluble in water.

ultimately, the purpose of the solvent is to "carry away" the dirt. but as i said, both organic solvents and detergent-based degreasers exploit this property... just in different ways.