How did Egypt become imperialized?

Answer 1

The term "imperialized" is confusing.

If you mean to ask, how did Egypt become an empire, referring to the various Egyptian such as Ramses II or the Fatimid Caliphate, the answer is typically strong governance and an outward looking military policy.

If you mean to ask, how did Egypt become colonized by the British Empire, the answer is that the British had superior military power and were able to force the Khedivites to concede defeat. The British then proceeded to use them as a puppet government to avoid massive spending on Egyptian infrastructure.