These files have something to do with python code analysis. More than that, probably. An example:
https://github.com/fastogt/qscintilla/blob/master/qsci/api/python/Python-3.4.api
These files belong either to Python, or to Qt ecosystem. The question would have been trivial; unfortunately the unhappy abbreviation disables the search completely.
What are these files and what does api mean in this particular context?
It's Scintilla code editing component: https://www.scintilla.org
The *.api files are language agnostic; they indeed feature in Qt apps, like Eric IDE, since Qt library incorporates Scintilla as QScintilla. So it has something to do with Python, too.