Storing the connection string in a configuration file gives you much more flexibility because you can simply edit the file and change the settings if you change or add a server.
Connection strings (and all other sensitive information) should be encrypted or hashed wherever possible. Storing a database connection string in plain-text (human readable) form can present a huge security vulnerability and should be avoided if possible.
By default, you can access the ConnectionStrings property almost identically to how you access AppSettings.