A few caveats before I begin. It's a shame to have to say this but there are actually contentious persons lurking on the web ... and pedants. So let me just state right up front I'm no Master of HistoryTM. I'm just a fellow who did a bit of research. There's always someone with more knowledge and long list of exceptions to general knowledge. Okay? This isn't that post. This is just a bit of information a referee may find useful. If you'd like to shed some authoritative light on any of the bullet points below, feel free to share.
Leather armor isn't soft or supple. It's hard as nails and effective protection. Those two qualities also mean it is stiff and unyielding. Leather armor or cuir bolloi was made by boiling a shaped leather piece in tallow or oil. It's closer in aspect to a piece of flexible sheet metal than a soft leather jherkin (which is how many neophyte players to the game picture it).
So, why leather armor if chain or plate was so much better? Well, leather was plentiful and inexpensive in relation to metal. It could be produced faster than metal armors.
Why not leather armor? Metal was better. So, if you were a big deal you would seek the best protection you could find. Metal armor was much more difficult to come by in terms of both availability and cost, so leather armor was better than nothing.
On a side note: I've been told by those who know better than I that leather armor was more effective than typically represented in the pre-1983 D&D rulebooks. Given I've heard the same about shields? I can buy that. I've no desire to change it because I'm of the opinion that, as a game-ism, it works just fine. But if you're a hardcore simulationist you may want to take a closer look at that aspect of your campaign.
So what was studded leather armor then? I've been told by Medieval wonks that this simply did not exist in the way described in some rules sets. It's possible the author was confusing it with jack or brigandine.