There are (at least) two ways of securing a wheel into a frame
. You seem to have a solid axe, with nuts that bolt the wheel in place. You have removed those nuts, but should have left them in place. The (principal) alternative is quick release, where a skewer goes through a hollow axle, and a lever is flipped to secure the wheel. As a broad statement, quality bikes tend to use quick release (QR), and cheaper ones solid axles.
Trainers need to accommodate both designs. Many basic trainers at least will simply take both - mine does. However, some trainers need different adaptors for each. Also, the lightweight QR skewer designs in widespread use don't take kindly to trainers, so you tend to get replacement ones for trainer use, which is why you got one with your trainer. (If you do have QR wheels, there's no need to swap back - it's fine to use a trainer skewer for road riding.)
I can't figure out whether your trainer will take your solid axle without adaptation but, if not, Paulatic has given you the (inexpensive) solution - swap the adaptors on the trainer. Either way, the QR you have is irrelevant for a solid-axle wheel. Put it away somewhere in case you buy a fancy bike some time.Edit: looking a bit more carefully, the nuts that Paulatic mentioned are meant to replace the wheel nuts that you removed. The reason will be that the new nuts are shaped to fit your trainer. As with the QR for the alternative kind of wheel, it would appear that there's no reason why, once you've swapped them, you can't leave the replacement nuts in place when road riding.
If you can track down a manual for your trainer to download on the Web somewhere, you may be able to get confirmation.