Where does the proxy reside?

It depends on whether you are referring to a server or host machine, an individual workstation and/or your browser settings. Proxy settings dictate how a standalone workstation is connected to the internet; however, most workstations that are not standalone rely on a dedicated server or host machine (which acts as an internet gateway) in order to access the internet.

Corporations use SSL Proxy settings to protect their corporate network as well as their intranet (i.e., internal network) from external threats that are most prevalent on the internet.

This is only a high-level description. More information about how servers are configured to connect to the internet, is available in the related link.