partially true.
splitting up into different databases doesn't do much for performance unless you can run each database on a different machine.
If you can't do that, splitting up is just making it harder on yourself.
That flatter the better only holds true for databases with proper locking mechanisms.
MySQL has a notoriously awfull locking mechanism. It locks tables instead of rows or records.
Hence, evey UPDATE/INSERT will lock the entire table, disallowing any other access to that table untill the query is finished.
For this reason alone, splitting up data into several tables can boost performance in situations where there are a lot of database inserts and updates.
