What it is
Look at any haute-horlogerie movement bridge through a loupe and you'll see the edge runs at a polished diagonal between the top surface and the side. That diagonal is the anglage: a chamfer cut at roughly 45 degrees and then mirror-polished. The technique has no functional purpose; it's purely a visual signal that the maker is willing to spend hours finishing surfaces no one will ever see in normal wear. Below haute-horlogerie tier, anglage is typically machine-cut and not polished; the bridge edge looks dull or industrial.
How it's done
The watchmaker uses a wood or hard-steel polishing stick charged with abrasive paste, drawn along the bridge edge at the chamfer angle. Each draw removes a tiny amount of metal; thousands of draws build up the polished surface. Inside corners (where two bridge edges meet at less than 90 degrees) require hand-cutting with a tiny chisel before polishing because no machine can reach them. A serious anglage job on a complex movement (chronograph, perpetual calendar) takes 8-30+ hours per watch.
What makes it haute
The inside angle is the differentiator. A machined bridge can have a flat anglage on straight edges, but inside angles (the V-shape where two anglaged surfaces meet) cannot be machined; they must be cut by hand. Patek Philippe, A. Lange & Söhne, Audemars Piguet, Vacheron Constantin, F.P. Journe, Greubel Forsey, Voutilainen all have hand-cut inside angles. Most other 'finished' watches have rounded inside angles where the machine couldn't reach.
What you pay for
Anglage is part of why CHF 50,000+ watches cost what they do. The labour is non-mechanisable: even with skilled watchmakers, a single bridge takes 1-3 hours to finish properly. A modern Patek 5170G chronograph movement has roughly 30 visible bridges, all anglaged; ~60 hours of finishing labour at watchmaker rates. See wiki: anglage.