The website you found pretty much covers it, so I am at a loss to understand what you want to know. You need to contact the relevant Ausländerbehörde (if you are not in Berlin, Google “Ausländerbehörde + name of the place“) as soon as possible and provide all the documentation listed.
Note that, as stated on the website,
An extension to a Schengen visa is only possible in exceptional cases where new facts and special reasons arise after entry.
The bit about late entry and unused stay is already stretching the rules, you are never entitled to an extension and certainly not to stay longer without having been formally allowed to do so. So you need to make an appointment/go to the foreigners' office and hand in your form soon enough to get an extension before August 30.
If you are not able to do this or the extension is refused, you have to leave. You can't simply stay in the Schengen area and count on your late entry to avoid any negative consequences.