They are suckers, you lucky person. I wish ours would do it.
They may be removed and grown on, but from what I hear, it can be difficult. You need to make sure that they have a reasonable set of feeding roots as well as the main rot from which they are shooting.
One word of warning, if yours is a grafted plant then the suckers will be from the rootstock and may well be some other kind of Daphne. Check that the leaves on the suckers are the same as the mother plant.