What is Stinespring Dilation?

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) $$\rho \mapsto \Lambda(\rho)$$.