Well the explicit aim is extinguishment. 
But besides that, there’s a few benefits I’ve found out, besides the benefits of a lower metabolism (with the caveat that I’m in no way medically trained, and this is just a dilettante practitioner’s researches):
For breathing, Oxygen specifically, is like fire. We need to breath rapidly to convert ATP into energy in a flight-or-fight situation, but that is also a destructive process.
By slowing down breathing considerably, we generate less oxidative stress. You know all those drugs and herbs etc. that’s promoting “Antioxidant properties”? Well, breathing less literally does the same thing, instantly, for free.
(Hyperoxia - being exposed to high levels of O2 (Wikipedia))
(Oxygen toxicity: cellular mechanisms in normobaric hyperoxia)
Then there’s your body’s entire O2 / CO2 sensitivity is a whole another thing. For example, trying to hold my breath right now in the morning randomly, I would gas out and suffocate under a minute easily. Easing into it in meditation, I’ve seen 3-5 minutes per breath easily.
The more you’re used to higher CO2 levels, the less anxious you get. This is a gross oversimplification of these recursive systems, but that’s the gist of it. By recursive, I mean that these systems are weird - just force yourself to breath faster and shallower, and your body starts to think you’re in danger, and raises stress levels to prepare for action.
Thus people who chronically overbreathe (low CO2, high O2) become more and more sensitive to small CO2 rises, which can trigger anxious, FoF responses more easily.
(Study shows how slow breathing induces tranquility)
Interestingly enough, Hyperventilation (fast, shallow breathing) causes cerebral vasoconstriction, reducing cerebral blood flow to decrease the oxygen supply in the brain.
I think this is because the body needs to protect the sensitive brain from overpumping it with O2. If O2 levels are high in the blood, the only way your body can handle this task is by reducing the blood flow to the brain.
So you’re simultaneously causing oxidative stress in the entire body, whilst depriving your brain of the blood it actually needs. 
Since breathing is so intricately linked to our FoF system, it bears to mention generally why having high anxiety / FoF activity is bad in the long run.
FoF mechanisms are a way for us to sprint for a short while to deal with a fast paced, short danger quickly. They’re a fallback survival mechanism. To give you the energy and strength you need to fight a predator or fly the scene, body shuts down a TON of non-essential mechanisms, like digestion, muscle building, auto immune activity, etc.
If you’re trying to prevent a nuclear fallout, you don’t have time to fit your tie, do you?
TL;DR breathing is like small explosions inside the body, like in the case of a combustion car engine. It’s absolutely necessary for the body to move, but being literal combustion explosions (in micro scale), you can see how it’s wise to regulate it.
As for Buddhistic guides on “Breathing”, really the only suttas we have are MN 118 (same trope repeated in MN 86, SN54.12, SN54.8, SN54.11) / SA 810. The 16 steps of breathing trope is probably the only actual practical guide you’ll find on breathing in the EBTs.
Visuddhimagga / Vimuttimagga might have some more details, I don’t know. They’re too dense for my smooth brain. 
Bk. Thanissaro’s With Each % Every Breath is an amazing analysis of practice, full of practical guides, you can see his actual experience in his words.
Bk. Analayo’s Praise for Mindfulness of Breathing touts itself as a practice guide, but I find it too academic compared to Thanissaro.
Analayo does stick to suttas more precisely, and Thanissaro relies obviously more on his own experience. For practice, I prefer the latter, but the former is still invaluable as an academic study.
) you’ll be able to tell what it is your body needs more.
