The primary driving force of protein folding is burying hydrophobic residues in the core of the structure away from water. Putting a protein in a denaturing solution, such as urea overwhelms this driving force through mechanisms which are not completely understood. The traditional method, involves chemically treating the protein to break disulfide bridges, performing a partial digestion of the protein to get smaller fragments, separating the fragments chromate graphically, then sequencing using an Edman degradation (which pulls off a single amino acid at a time). You then align the overlapping fragments to get the full sequence. A variation on that process has been semi-automated in the form of protein sequencer devices. These days, the dominant method is probably one of the mass spectrographic approaches.
https://en.wikipedia.org/wiki/Protein_sequencing