Okay, breathe. From what you say, it may not actually be NEC, but if he's at a little unit makes sense to transfer to a bigger place.
NEC is something that can happen in preterm babies, and is more common if they have IUGR. Roughly, the theory goes that if the baby isn't getting enough supply of oxygen/nutrients, because the placenta is failing, then the body concentrates on supplying the vital bits like the brain, so less important bits like the gut can take a hit. When IUGR premmies are born, knowing that this is a risk, usually feeding is started very slowly, to avoid upsetting the gut, with a drip to provide the rest. Where possible, breast milk is used as this is gentler on the gut - your SIL may well be expressing. When first trying to get feeding going it can be very stop-start-stop. Before each tiny feed they'll check what's left in the tummy. If the last feed is still there, they usually won't give more. If it's tinged green (the natural colour of stuff in the gut a bit past the stomach), they won't feed, and will rest the tummy. Most of the time the green settles as the gut gets used to doing its thing. But sometimes it can be an early sign of NEC. And because of that, small units tend to send babies to big units as soon as the green isn't settling, even though lots of them will be fine, because for the few who do have NEC it's better to be safely transferred.
In NEC, basically long stretches of the gut don't work right. They get sore/inflamed. This can make the gut walls become leaky. Which is why the antibiotics, as that way any bugs leaking out can be killed off. The most important treatment is to stop feeding, as that gives the gut a rest. Usually the feeds would be stopped for at least a few days, which means that just a drip might not be enough. There are special sorts of medical nutrition that can be given IV, but these may not be available at the smaller units. It's not unusual for the need for that type of feed to be a reason for transfer. Sometimes NEC can be more serious, needing surgery. But not usually.
Hope that makes sense. Oh, and it terms of transfers...in a daft sort of way it's good. If the transfer had happened really fast, that would imply they were really worried about the baby. Being a lower priority transfer implies that they're less worried.