A trip between two Schengen countries should basically equal a domestic trip, so no passport checks for example (other than if needed for ID purposes such as at check-in).
If you need a visa you will also apply for a Schengen-visa which is valid for all Schengen countries.

At FRA and other European airports they need to both have a Schengen transit area and a non-Schengen transit area.
The non-Schengen area is there so people travelling for example between the US and the Middle-east should not have to enter the Schengen area (i.e. enter Germany).
However for a passenger travelling for example from the US to Paris/France via FRA the passport control will take place at the first point of entry into the Schengen area. So as Frankfurt-Paris is a Schengen flight (both countries are part of Schengen) the passenger will enter the Schengen area in Frankfurt and passport check will take place in Frankfurt.

May sound a bit complicated, but it is very convinient for people travelling frequently within western Europe as it means less hassle and no passport checks.