Webex offers a few different Webex Edge options. They can be confusing. I will try to outline what they are and what do they do:

Webex Edge is a great cost saving feature for customers who utilize Webex Meetings and CUCM...
This allows Webex Meetings to route PSTN calls through the internet and leverage on-premise CUCM for Webex Meetings Outbound/Call-Back feature - saving $$ on the Webex PSTN costs..

Here is a webex.com article on Webex Edge Audio.

Webex Edge Connect allows a customer to have a private point-to-point link between their network and Cisco Webex (Meetings and Calling) Cloud. This allows to by-pass public internet - thus guaranteeing bandwidth and quality of service (QoS).
Here is a webex.com article on Webex Edge Audio.

Webex Edge Video Mesh allows local (on-prem) media processing for cloud based media services thus improving the customer experience for on-prem users.
It is a software which is installed on-prem which is cloud managed by the Webex Control Hub.
​Here is a cisco.com article covering all of Webex Edge products.