WinSCP suddenly opens with two local panels with seemingly no option to open a connection to my server. What do I do?

You have probably by mistake disabled automatic popup of the Login dialog on startup.

To restore it:

• go to Preferences (Options > Preferences),
• select Environment > Window page,
• and check Show login dialog on startup and when the last session is closed.

Alternatively, you can just open the Login dialog manually.

You can do that by going to Tabs > New Tab [> Remote] (or simply by clicking New Tab tab).