It might be easier to explain backwards than forwards:
legal protections and secure rights.
Okay, this bit's easy enough, protections that are provided by law and rights that are secure and hence that people can feel secure in.
on the edge of legal protections and secure rights.
We have a figurative use of the spatial meaning of edge. We can picture someone who enjoys legal protections as being "within" them. We can picture someone does not who enjoy legal protections as being "outside" of them. Someone who has an incomplete, dubious or unreliable access to legal protections, we can hence consider as "on the edge of" them. Likewise with secure rights.
place economically vulnerable populations on the edge of legal protections and secure rights.
Continuing the figurative use of spatial terms (or rather starting it, since we've moved backwards to this point), to place someone on the edge means to make it such that they are now positioned there. Hence here, this means that those economically vulnerable populations are caused to be put into a situation where their legal protections are incomplete or unreliable.
And as such, "to place [someone] on the edge of [something]" means here, to make it so that [someone] is in a position in which they are neither fully within or fully outside of [something].