1) Router administrator negotiates an HTTPS or SSH session with a router or other hardware
2) Attacker is either listening passively or is a man in the middle (via ARP poisoning or what have you). Because they have the private key, they can advertise themselves as being the router without raising the alarm with your SSH client or browser
3) You provide credentials to the router (or MITM). The credentials are logged by the attacker
4) You proceed to do whatever you intended to do in the router's configuration, and log out.
5) Some time later, the attacker logs into the router as you, and makes nefarious changes to the router configuration (such as uploading compromised firmware which logs traffic, or has a backdoor, etc). Any changes done look like they've been done by the router administrator.
I don't know how likely this is in a work scenario though; I haven't searched the database for common mid-level to enterprise routers/remotely configurable switches. More than likely, in a work situation, you'd be using hardware which generates a key pair upon initial configuration. The scenario above is more likely to apply to SOHO, or to consumer wireless hardware in the home.