They do a number of things. The first is they trap like crazy and basically force teams to dump and chase, with that, they have guys like Ehlers and Connor who are absurdly good at breaking up plays and creating rush chances. So they break up controlled possession with the best of the league through that trap and they counter attack very well out of it. This forces a lot of teams to dump and chase, which today there are not many teams that do that well. They also are a heavy pressure team on offense... they are very willing to hit, grind, and cycle you to death. They may not create a ton of shots or chances, but they get defensemen a little wary of physical play... and they make teams just flat out work to get possession. In their own zone, they push teams to the outside and collapse very simply into the middle of the ice. Helly doesn't have the greatest rebound control for an elite goalie... so they designed their scheme around that. They'll fall back with 3 players in the high danger area ready to pickup rebounds and make simple clears where they are just fine (when they have the lead) to get the redline and dump. If you do get rebound chances against, they tend to be at least contested. They'll let you play outside the dots all you want... but odds are you won't get more than one or two shots per entry if you attack like that. They force you to try to draw guys out of that area to get sustained pressure... where they are a very disciplined group defensively.
It is pretty clear which teams just destroy Winnipeg and it is the teams that are just more than willing to play that dump and chase, grind it out sort of games. Flyers, Vegas, Stars...even Calgary. If you're willing to slog out a game, Winnipeg has issues tweaking their system to work against that. If you only want to play high flying, pure possession hockey... they can just bore you to death.