I've consulted Nielsen and Chuang to understand the Stinespring Dilation, but wasn't able to find anything useful. How does this operation relate to partial trace, Kraus operators, and purification?
Not exactly sure what you find confusing, but the ultimate need for Stinespring dilation theorem is that in quantum mechanics the dynamics is in general defined by a completely positive trace preserving map (CPTP) $\rho \mapsto \Lambda(\rho)$.
Now, we have a belief (rightly or wrongly) that all there is is a unitary evolution governed by Schrodinger's equation. Stinespring dilation theorem is a way of stating this desire from a mathematical point of view. It guarantees that as long as I see the evolution of my density matrix as governed by a positive map (in physics we require trace preserving), this CPTP map can ultimately be lifted to a unitary operation on a higher dimensional Hilbert, where we can simply think of unitary evolution (purification).
Given that this is the case, how do we go back to the lower dimensional space? The prescription is that we trace over the system we appended to get unitary dynamics and then the dynamics in the lower Hilbert space is governed by a set of Kraus operators. This procedure is guaranteed by Stinespring dilation theorem.