I’ve always had doubts about war games, in the lore they say they are simulations, but if we see our spartan die, is it really that they are connected in a pod in a Rick and morty style simulation or something?
Ive also heard heard that the War Game spartans are just running a training simulation to further enhance and perfect their skills. Now the exact process of how this works I really don’t know, but I think (personal opinion) that its more in a neurological way, something where the spartans can sit or lay down and connect themselves or with some help on to some pods maybe and their connected in a neurological way where their conciseness is transferred to a highly realistic training simulation, where they actually feel like their inside of their bodies, and probably they might feel some low voltage electrical stuns when they get shot or take damage, resulting in the brain to exercise in decision making, combat effectiveness, strategies, teamwork, weapon handling, communication, and many other things. When they decide to finish training they just disconnect and leave and probably hit the gym or something as they (you can say) trained mentally or practiced. Basically that’s my take and my kinda logical way of thinking about it, I also took some terms/ideas from other games so it can seem as logical as possible.
The training in Halo 4 and Halo 5 all happens inside the UNSC Infinity. Basically, it’s a hyper-realistic simulation, but they are not just hooked up and partaking video game style. They are all actually firing weapons and moving and punching. The weapons are modified to be non-lethal with bullets and alien weapons using plasma and such are toned-down or are replaced with realistic holograms. Everything you see in War Games is just holograms. Reverse engineered Forerunner tech allows this.
As far as I know there is no lore explaining how your spartan actually “dies” in the simulation, but it’s likely that they experience actual pain equal to that of the traumatic event and thus are rendered unconscious and are teleported or moved. There was one time a Spartan actually died during the simulation because he was killed by a traitor who was using an actual knife and stabbed him in the neck. So the combat is pretty real.
The Mjolnir is able to fully create a virtual environment around the spartans (An AI used this to turn Black Team on each other in a comic).
Between that and an X-men style “Danger room” crossed with star trek’s “Holodeck”, and possibly a localized teleportation grid for re-spawns, The spartans could be actually facing off against each other and virtual Covenant and Prometheans, though their projectiles are probably virtual to limit the risk of loss of assets.
This is all technically within humanity’s ability to create, especially since they’ve been studying Forerunner tech for several years.
They simply die and then respawn in the simulation 
on the infinity they are able to run a simulation to help train the spartans. thats also why there are maps that wouldn’t actually exist. If you have ever played the titanfall 2 campaign where you start off in the virtual world it would work like that.