I'm trying to understand the deeper philosophy here. How do Brahman, Avidya, and the Unmanifest all relate to each other?

Chapter 50 details a complex relationship between these concepts. Brahman in its pure form is described as separate from the 'expressed' Brahman. It is Avidya, or ignorance, that qualifies Brahman, making it 'shabal' and allowing it to be expressed as 'Sat' (Existence). This expressible Brahman, graspable by the intellect, then gives rise to the 'Unmanifest' (Avyakta). This Unmanifest is also identified as the 'causal body' of the soul, which is itself caused by the ignorance of the Soul. Therefore, Avidya acts as the qualifying principle that veils pure Brahman, allowing for the emergence of the expressed Brahman, from which the Unmanifest and the entire worldly illusion subsequently unfold.