"Admitted to the degree of" meaning?

what does “admitted to the degree of” that you are starting a degree or it have finished a degree?

Probably starting, since the names says “admitted to” instead of simply “degree of” or “awarded degree of” or something of the like.

What’s the context for this?

It’s an Anglo way of saying what Americans mean by Awarded the degree. Americans tend to think of a degree as a thing, where everyone else thinks of it as a level.

Think of a degree as analogous to a floor in a building. When you step off the elevator, you are admitted to the floor. You’re not awarded the floor.