purpose of push-in type electrical connectors?

Quick question.

I will post a pic if need be.

I have a wire coming off the fan that goes into an electrical connector (the push connection type).

Well the wire is fraying and coming out of this connector.

Does this connector serve any purpose—like is there a sensor of any sort or is it merely a connector?

I was thinking about cutting both sides and jumping them together with a butt connector—but ONLY if the push-in connector served no functional purpose.

Any reply appreciated! Thanks!

-Mike