That happens whenever you try to play ESF in singleplayer mode, because ESF wasn't made to work in singleplayer mode.
This method is a little longer than the -console command in the link target, but I've never found a situation yet in which it didn't work.
In your ESF directory (should be something like C:\Sierra\Half-Life\esf), there is a file called config.cfg. Open it with notepad. This file is run by ESF each time you start a game, and sets all your key binds.
Somewhere in the file (doesn't really matter where) insert the following lines:
bind "~" "toggleconsole"
bind "`" "toggleconsole"
Save the file, and now the ~ key should bring the console down! ^.^