No offense, but what's the point of a psionic class in D&D? What would they do that isn't covered by any of the spell schools? There's Divination spells for telepathy, Enchantment spells for controlling people's minds, Transmutation and Evocation for telekinesis, and so on.
(this post is a bit of a generalization of all pre-3E psionic powers, as they changed quite a bit depending on the era)
Well, for starters, it's
not magic. Which is incredibly important in settings like Forgotten Realms.
In FR, mages are technically a kind of cleric really. They all get their power from some deity(most of them Mystra,) and Mystra(and her various incarnations) are the reason the weave exists and magic works at all.
Without the weave, magic that depends on it just stops working completely.
So, the Spellplague, Time of Troubles(dead-magic zones), etc?
Yeah, psionic powers kept working just fine because it turns out that psionics are vastly more powerful than silly not-sorta-clerics. Standard tactics to suppress or nullify frilly dress wearers didn't work on psions.
But to answer your question in proper, refer to the Complete Psionics Handbook. It actually covers this in one of the very first pages:
Additionally, psionic combat was its own separate thing back in AD&D. Your guy with psionic powers could be having his own separate mind duel with the Big Bad Guy while your regular dudes were having a physical battle with the Big Bad Guy.
How did it work? Here's an explanatory video that walks you through it:
Psions also used a different resource mechanic than standard spell slots -- it was more akin to mana that slowly recovered. You could also burn HP for extra psi points depending on the exact edition/supplement you were using.