Checking out a license remotely requires VPN software. With the VPN, you're making your computer believe it's on your work network, and the network behaves similarly. You can see network drives, access internal database servers, etc. If you VPN into work before opening ArcGIS Pro, only then can it see your ArcGIS License Administrator and verify your named user license.
Now, I'm no professional when it comes to networks, ports, tunneling, handshakes, and all that; there is perhaps some setup that allows remote use of Pro, but I haven't figured it out yet.
I did log a ticket with Esri Support, and that's what I took from it.