I see there is a freeform forms fieldtype available after the plugin was installed. What exactly does this field do and how to render it correctly (within matrix)?
From the Freeform docs:
The Freeform Form (or Submissions) fieldtype lets the user assign any Freeform form (or form submissions) to any element: a section entry, categories, assets, etc.
In a nutshell, this means you can embed forms to any entries you have, like so:
This section within the docs shows you how to correctly code your templates.