The biology of outsiders simply requires them to breathe.
The notion that outsiders need to breathe, but not eat or sleep, was introduced in the D&D 3.5 Monster Manual without any lore explanation. It didn't appear in D&D 3.0, and did not appear in earlier or later editions (which lacked an outsider type).
Prior to D&D 3.0, which created the outsider type, there wasn't much of a sense that creatures of the Outer Planes formed a unified group with similar physical characteristics. The D&D 3.0 Monster Manual didn't have any specific rules on whether each creature type needs to eat, sleep, or breathe, and there generally aren't creature-specific rules for those activities either. The closest is the 3.0 Dungeon Master's Guide p.116, which notes that monsters in a dungeon generally need to eat, drink, breathe and sleep.
The D&D 3.5 Monster Manual, for the first time, formally defines whether each creature type needs to eat, breathe, and sleep. This is obvious for some types (humanoids are living and thus do all three, constructs and undead are non-living and don't do any), but outsiders are in an odd situation as a kind of supernatural being for which there is no pseudoscientific or well-defined lore reason. The authors simply had to pick an answer for each of the three categories.
D&D 4e no longer has the outsider type (although there are some subtype-specific definitions, like that demons both breathe and eat) and D&D 5e no longer has type-based rules. This suggests that most of our lore comes specifically from D&D 3.5 sourcebooks (2003-2007), which don't define outsider physiology in sufficient detail.
Fiendish Codex: Hordes of the Abyss p.7 describes the physiology of demons in particular. It explains that they possess lungs and have a physiological need to breathe, and can breathe even poisonous fumes with ease. However, they do not have the same needs as a creature of the Material Plane, and do not require food or sleep. No reason is given for why they need to breathe, except that the simply have a physiological need to do so.
A possible element of pre-3e lore support is the Swallowed Void, the 68th layer of the Abyss (from AD&D 2e Planescape work Planes of Chaos), which is an airless void, and no demons live there. This might imply that even demons can't survive in a vacuum, otherwise some demon lord would have claimed it.
I suspect that the 3.5 definition is based on practical concerns. If outsiders generally did not breathe air, it would be difficult to explain why the Outer Planes are full of air (which they were in earlier editions, and generally have to be so that PCs can easily explore them), or why they can speak languages. It's also combat-relevant, e.g. gas-based spells like cloudkill, whereas it doesn't matter so much whether your creature needs to eat or sleep, because going without these for the duration of a combat is less urgent.
