To deputize is to appoint someone as a substitute, like a police deputy authorizing a civilian to make an arrest.
The original meaning refers to when a deputy would give some of their power to non-police officers. When deputized, you took on some of the power of a deputy to help the real deputy. Similarly, deputizing can refer to making anyone a substitute. A teacher can deputize a student by putting them in charge of the class. A boss can deputize an employee by asking them to organize a meeting. Deputizing empowers someone to do a job.