You need to scrape the "+" and "-" signs off the XT-60 connector for it to fit. The fit is very tight, but a drop of CA glue inside will not hurt.
The "shroud" part goes inside the hole in the metal ATX case and the ring secures it in place from the inside. Use provided "wrench" tool to tighten the ring if needed.
On some supplies, the multiple wires of each rail (and ground) are soldered together through a large hole in the PCB. You can use those holes to solder proper 12 AWG silicone wires (or even thicker) to the board and route them to the XT60 connector.
If the wires are soldered separately, you will need to solder multiple wires together to a single XT-60 pin not to waste any power and provide all the current you need.
Watch out for modern supplies which have voltage sensing wires on a subset of rails (look for multiple wires terminated in a single ATX plug pin). Those need to be connected to the respective rails after removing the original cables, or the PSU will not start.
Last but not least, remember about PS_ON!:)