Advancing, in this use, is the fact of growing older. We most often meet it in set phrases: advancing years, advancing age, with advancing age. It is a gentle, slightly formal way of talking about getting old, without saying “old” outright. Doctors, writers, and polite speakers use it. Though actually, the same form advancing is also the ordinary -ing form of advance — an army advancing, technology advancing. This entry is only the “growing older” meaning.

年紀漸長；年事漸高（多見於 advancing years / advancing age）