It will be asking you for the Admin account username and password.
You won't need to use the MySQL username/password through the web pages. You will only need to use those with PHPMyAdmin to manually edit or backup your database.
What I would strongly suggest is that you consider manually installing phpBB without using Softaculous. This requires you to do a little more, but in the long run you will have a much more adaptable site. This auto-installation systems, like Softaculous, are good in principal. However, in practice, they usually cause more hassles in the long run.
I've just had a look around LunarForums for a more up to date tutorial on installing phpBB that the one I previously wrote, but didn't turn anything up. You can take a look at that to get the general idea of what sort of things you would have to do to install it yourself, without Softaculous. However, things are somewhat different (and in general easier) these days.
I'll perhaps write an updated tutorial for phpBB soon. In the mean time, the old one can be found at: http://www.lunarforums.com/web_hosting_tutorials_faqs_and_resources/how_to_manually_install_phpbb-t9761.0.html
Additionally, there has always been good help and advice on the phpBB website http://www.phpbb.com/