What is fix geometry actually doing in QGIS?

As per the OGC Simple Feature Access specification, geometries need to follow the OpenGIS compliance (see the PostGIS docs for quick reference), where (listing only the most prominent predicates)

• Points are considered inherently valid
• LineStrings are valid if they are simple, meaning that they don't pass an inner vertex twice
• Polygons are valid if their linear components are simple and none of their rings cross
• MultiPoints are valid if no coordinate pair is present twice
• MultiLineStrings are simple if all their components are simple, and common vertices only touch
• MultiPolygons are valid if all their components are valid and do not overlap

In order to do so, the internally called GEOS modules will (listing only the most prominent operations)

• correctly node a LineString and remove consecutive duplicated vertices
• correctly node the linear components of a Polygon and iteratively rebuild all possible valid areas from them
• do the above for all parts of their Multi geometries

where noding means breaking apart a linear component at non-consecutive duplicated vertices.

In this context, 'fixing' geometries rather means creating the maximal set of their valid and/or simple parts, and may result in a collection of geometries of different dimensionality.