When people talk about machine finishing, what they're typically referring to is the anglage on the bridges. Machine finished anglage will often look flat and may have slight vertical striations visible left over from the machining process. What you ideally want to see for hand-finishing are round, wide, perfectly polished bevels, preferably with interior angles. Interior angles are notable because those can only be finished by hand, and are supposedly more difficult to finish as well.
To show some examples with pictures....
Machine finished anglage on a simple Patek. Notice the vertical striations? I've seen photos of examples with what appeared to be better quality anglage though, so maybe this varies on a case by case basis.
Machine finished anglage on a Moser:
Royal Oak anglage. I don't see clear machining marks from this photo, so I may consider it better than the last two examples.
And for some examples of hand-finished anglage...
Parmigiani Hebdomadaire. An unusually large number of interior angles, but would have been nicer if the anglage was a little thicker and rounder.
Credor Eichi II. Incredibly wide, rounded bevels.
Duffour Simplicity. Probably considered the gold standard for what expert-level anglage should look like.
Romain Gauthier Logical One. Very wide, rounded anglage with many interior angles.