On the physics:
If you take a sub, scale all its dimensions up in the same way, the resulting bigger sub won't be able to dive as deep as the original one. That is because you are decreasing the radius of the hull as you are increasing its size, and it is the curvature of the hull that allows it to withstand the water pressure.
Obviously, you can still increase the size of a sub by simply joining several subs together. Likely, those bound-together pressure cylinders will be enclosed within a thin, non-pressure withstanding hull for hydro-dynamic reasons. Each pressure cylinder, however, will have a rather small diameter, to allow for large diving depths.
How to explain gigantic subs:
Well, you obviously need a reason for large subs. As a matter of fact, subs are relatively poor at actually controlling an area (or water volume). To do this, it might be necessary to use a large fleet of small subs. However, small subs cannot use nuclear reactors, as those have a certain minimal size. So, the small sub fleet would have to do with battery power. But how can you reload your batteries when you are on the other side of the planet, and want to avoid surfacing for whatever reasons? Enter the big sub carrying sub!
It would work a bit like aircraft carries today: You have one really big vessel, which contains at least a dozen small vessels, which are the big vessel's eyes and arms. These small vessels patrol the area that is to be controlled, and they strike against any enemy they find within their action radius. Once their batteries are down, they return to their carrier for a battery charge, remunitioning, and some fresh supplies.
The carrier sub itself would be huge by todays standards. It would likely contain several powerful nuclear reactors, large ammunition storage areas, and have several places for the small subs to dock. The parts that need to be at atmospheric pressure would all be contained within a number of long cylinders, each no more than five meters in diameter for structural efficiency. This pack-of-thin-cylinders design would go quite well with its purpose of being a carrier of many small subs, I think.
The big sub itself would not have much defensive capabilities itself, relying on its fleet of small subs to defend it, just like aircraft carriers rely on their air force to defend them. The big carrier sub might contain a few big offensive missile type weapons, though; weapons larger than the small subs can carry.
Such a carrier sub would really be able to control a certain volume of the sea, which is hard to do with the modern subs which are mostly built for offensive strikes on the surface. The modern subs are built that way, because we humans are most interested in controlling the earths surface, for obvious reasons. A civilization that mostly lives underwater would have much different priorities: They would desire to control either water volume, or areas on the bottom of the oceans, so I think, big carrier subs would suit them well.