After you have enabled TCP/IP (see Chage's answer), you will need to connect to your SQL Server Instance.
When you were installing it, it may have asked if you if you wanted to install it as a "Default Instance" or a "Named Instance".
Chances are, if it's SQL Express it's a named instance, and the default instance name is "SQLEXPRESS".
The way to connect to it, is to specify the server name as SERVER\INSTANCE, so in your case, "localhost\SQLEXPRESS". If you are using the "Default" instance, you just use the server name (drop the \INSTANCE) part.
The connection you are seeing to .\SQLEXPRESS is the connection to your instance. If you are using Integrated Authentication (it should also have asked you this during installation), then you will need to give the account that LINQPad is runing under permissions. This is another question alltogether ;) If you're running LINQPad under your current user then you should be OK.