It’s hard to say. There are game logic reasons for these weapons, (i.e. gameplay balance, fun factor) and then there are lore reasons for these weapons. The two often contradict each other.
Let’s get this out of the way first: the DMR is a marksman rifle. It’s meant to be used by the squad marksman. The US Army does this in real life as well: there are one or two shooters within the squad who carry a longer range, slower firing battle rifle that’s designed to hit targets with precision.
Then we’ve got the AR and BR. In Halo 2, there was no AR. The BR, while replacing the CE Pistol for gameplay purposes, replaced the AR for in-universe lore purposes. It was essentially a new service rifle. Or so we thought… in Halo 3, the AR makes a triumphant return. The gameplay (aka real) reason this change was made was because Bungie preferred the balance of spawning with a mid-range rifle as opposed to a weak SMG. They saw how popular BR starts were in Halo 2 and wanted to make non-BR starts a little more popular. They also wanted to bring back an iconic Halo weapon, which despite being pretty mediocre in multiplayer, shreds pretty well in single player.
The lore/canon reasoning gets a little confusing here. Unlike, say, the US Army, where the M4 carbine is essentially the standard issue rifle, the Halo universe now effectively has two: the AR and BR. It’s hard to say which one is “standard” because so many sources, from the Halo Encyclopedia to the books to the games, seem to contradict each other. It’s pretty likely that we can assume the MA5 is the “standard issue” rifle of the UNSC, aka the rifle each UNSC Army or Marine trooper would be issued after basic training. However, we can also assume that the BR is slowly superseding the AR in use when available. We know this because of how popular it is as a service weapon, how common it is in Halo 2 and how it seems to exist in even numbers with the AR in all future Halo games.
You might argue that either rifle could be the standard. It’s a pretty extreme example, but let’s compare it to the way the US Army issues optics. The Army has two primary optics that it uses: the Aimpoint and the ACOG. The Aimpoint has no zoom; it’s a simple and extremely efficient red dot reflex sight, easy and fast to use, which makes it lethal. The Army also commonly issues the ACOG, a 4X zoom optic that gives enhanced accuracy over range at the cost of target acquisition speed.
While the Army issues both optics, the proportion of each is distributed based on a number of factors. Different units prefer more riflemen to have the Aimpoint; others prefer them to all have an ACOG. If you’re deploying to a dense, urban, battlefield, you might prefer an Aimpoint; if you’re in the mountains of Afghanistan, you might prefer the zoom of an ACOG.
You could compare these optics to the way the UNSC issues rifles. Sometimes, the AR is a preferable choice. It has a higher rate of fire and it’s an all-around workhorse, much like the US Army’s Aimpoint optic. On the other hand, sometimes the BR is preferred. It’s more precise at range, and it still offers a decent rate of fire.
The DMR is a specialist weapon. It’s a marksman rifle through-and-through. The AR and BR, on the other hand, are more numerous and can be used somewhat interchangeably.