On a regulator first stage with swivels, some of the low pressure ports are mounted on a separate part of the regulator body that can be turned. When hoses are attached to this part, such as your two second stages, the hoses can move a bit if they need to to give you more flexibility. For example, if you turn your head away from the side of your second stage, the first stage can swivel a bit to give you some more hose. That flexibility is the advantage of a swivel. The disadvantage of a swivel is that it is one more failure point. There has to be o rings in the swivel to keep it from leaking, and if one fails, you will get a bad air leak at the swivel point. That is why some cave divers won't use swiveling regulators. I have never heard of this actually happening, but it could. For that reason, I don't use swiveling regulators in caves. I have Scubapro Mk 16's, a non swiveling regulator first stage, for that. I also have Scubapro Mk 20's that do swivel for open water. I don't see that a swivel gives any real advantage, so prefer the Mk16's for the most part. The Mk 16 is a good regulator, not too expensive, and breathes just as good as the Mk 20. It is a Rodales' Scuba Diver "Testers Choice" because of it's breathing characteristics, and a "best buy" because of its relatively low price.
Happy researching!
Dave