Go to MP in the mods on the SP site, and click download for PC/Mac, and continue clicking download on the various pages that open. When the white download window opens, it should show open with SteamClientBootStrapper. Click OK, then wait while it downloads!

How to connect (borrowed from MP instructions):

To connect to a server, go into the designer, hit the new "MP" button, type in the server IP address, and hit join.
If this does not say you joined, or failed to connect, then you may be experiencing a bug, open the dev console with the backtick key, usually under esc, and type SetName [username] (not required, but recommended to ensure you don't show up as LazyNullName), and then Connect [IP]. If that still doesn’t work, you may need to restart SP and click join again.
When a checkbox is blank, that means UNchecked, a checkbox with black square in it means CHECKED.

To use the in-game "chat", open the dev console, and type
say "whatever you want to say&quot (for example, ‘say “something” ’);
Double quotes are required if you have spaces in whatever you want to say.

To make your own nametag visible, open the dev console and use the ToggleNameTag command.
Same thing to hide it again.
But if that doesn’t work, then
To host a server, look what the SP MP mod page says.