The JuMP docs now have a simple Dualization tutorial: https://jump.dev/JuMP.jl/dev/tutorials/conic/dualization/
But a deeper discussion on the primal/dual formulations and specific solvers is a but out of scope. However, the docs here can go into the weeds and explain the different internal formulations.
See jump-dev/JuMP.jl#3408
See jump-dev/JuMP.jl#3416