I have a rose from a florist that has started putting out leaves from where the thorns were removed. Is there any way I can "root" this rose and keep it growing or is this just a fluke? Thanks!
Not a fluke at all. Florist roses are "juiced": so pumped full of nutrients and grown in absolutely perfect conditions, that they will put out new foliage due to the massive amounts of nutrients they've been able to store in their stems due to those perfect conditions (rose storehouses are mainly their stems, unlike, say, a potato that forms tubers).
So, you can try rooting it, yes, it might take, but keep in mind that many florist roses are duds as garden roses. Some are fabulous garden roses, many are not. Google up "how to root a rose cutting", lots of advice and different techniques to try.
There is an older thread on the Propagation forum where a lady successfully rooted several. She posted blooms after it was mature. Go there and do a search.
The new leaves are coming from the base of old leaves. There are no leaf buds where the thorns were.