It’s a rather complicated and controversial area, but I will attempt to give a very concise overview (as far as I have understood it).
Contextually, Yogacarin and tathagatagarbha literature was in part a response to the way that some monks had understood the mahayana prajnaparamita literature. Some of this literature can be read as articulating a kind of illusionism in which the world is simply and straightforwardly empty like an illusion is empty, and nothing else. There is literally nothing at all behind the illusory appearance of a world. The problem with this view, as the Chan philosopher Tsung-mi writes, is that
If the mind and its objects are both non-existent, then who is it that knows they do not exist? Again, if there are no real things whatsoever, then on the basis of what are the illusions made to appear?… there has never been a case of the illusory things in the world before us being able to arise without being based on something real.
Indeed, one of the central texts articulating the ‘buddha-nature’ perspective (the Ratnagotravibhaga) presents itself as a “corrective” to those who have been waylaid by this negative or nihilistic understanding of emptiness.
In terms of arguments for the ‘buddha-nature’ view, much of it is based on scriptural authority (taken from canonical sources).
For Yogacarins there is an early Buddhist text called the Culasunnata-sutta, which speaks of an emptiness in which there is something left-over or that “remains” (a mysteriously affirmative form of emptiness that follows after the experience of negative emptiness).
Another textual precedent for the ‘buddha-nature’ (tathagatagarbha) schools is taken from the Anguttara Nikaya, where it says: “Luminous, monks, is the mind. And it is freed from incoming defilements” - the implication being that the mind is fundamentally free (luminous), but is covered over by the defilements of conceptual thought.
A general scriptural precedent for their view comes from the Samyutta Nikaya, which teaches that ‘reality’ (or nirvana as articulated in positive language) is “unborn, unproduced, uncreated, unformed”; without which there could be no release from the “born, the produced, the created, the formed” (Udana 8.1. and 8.3.). For the ‘buddha-nature’ school, this unborn, uncreated reality (variously called dharmakaya, dharmata, tathagatagarbha, tathata or suchness) can only be revealed to a mind that has cleansed itself of all conceptual defilements, and so is ‘empty’ in that sense. But it is an emptiness that makes the luminosity of the uncreated dharmakaya possible, and so is ultimately non-empty (or beyond emptiness and non-emptiness: i.e. ineffable).