Could I have my php scripts on server A and connect to the MySQL database on server B?
If yes, how it would be done? Thanks in advance
Yes.
The same way you access the localhost on the same server, you change the database host to the external one. This is more a configuration issue, you need to grant your database user remote access to your MySQL, you also need to make sure your firewall allows connections on the MySQL port.
Example on Debian: http://www.debianhelp.co.uk/remotemysql.htm