I’m assuming they’re like any other engine and need to mix air with the fuel in order to work, so where does all the air come from?
The Diesel engines are only run when surfaced, or else just submerged by means of a snorkel (or Schnorkel). The engines provide surface propulsion and charge the batteries. The subs run on batteries when they are submerged.
Subs typically run on the surface while using the engines. Sometimes they can cruise just under the surface using a snorkle. To dive they need to engage the electric motors running from batteries that were charged while running on the surface.
And when the snorkel was covered by a wave, and the floating ball valve cut off the air flow temporarily, the crew could suffer ruptured eardrums as the diesels drew in the air already in the sub.
It is worth noting that for some reason, the engines feed off the air in the crew compartment. So when you are running the diesels off the snorkel, you get a heck of a breeze in the crew compartment. If you have a fire on a sub, you surface and start the engines. They suck the smoke out so you can see what you’re doing.
Why would the engine’s air intake be connected to the interior air space?
They did that in Alistair MacLean’s Ice Station Zebra, so I was sure it was a crock. 
Less chance of the air flow being blocked or restricted.
The Swedes have an attack sub, the HMS Gotland, which uses tanks of liquid O2 to run the engines while underwater. They spent two years running circles around the US Navy off the West Coast as part of an ASW training program.
Surely, if war ever finally breaks out between the US and the Swedes, we’ll get a heck of a bloody nose. 
Yeah, we’d be borked, borked, borked.
http://roflrazzi.files.wordpress.com/2008/10/beaker-oh-shit.jpg?w=500&h=374
(maybe not work safe due to language)
Fixed the linky for you. And damn, it must take a lot of faith in German/Swedish engineering to volunteer to be sealed in steel can underwater with a whole lotta LOX. Yeesh!
What kind of weenies have their sub ferried to it’s base of operations?
Well, with fuel prices what they are, I wonder what kind of mileage that sub gets on its own?
What about GRAVLAX?
Sounds like a Led Zeppelin parody.
From what I’ve read about World War II US fleet submarines, there was a very large induction valve in the hull that was opened while running on the surface under diesel power.
The interior space acts as a “free” water separator. Sucking water into a diesel engine can be very destructive. Due to the high compression ratios, it doesn’t take much water to hydraulically lock the engine. Even without a complete hydraulic lock, the water can raise the compression enough that combustion pressures exceed design limits, blowing the head gaskets at best, or causing failure of the head bolts/studs at worst.
Additionally, without using the hull volume, the snorkel would need to handle the peak induction flow rate with minimal head loss. With the large hull volume, it only needs to handle the average rate.
Finally, since stealth is essential in sub operations, reducing the induction noise by any available methode is desireable.
And no bagels or cream cheese!
It also allows for continued engine operation when the snorkel head valve momentarily closes due to immersion by waves. The hull’s interior volume is used as a reservoir which the engine can draw on, creating a (hopefully) small vacuum until the valve re-opens.