The new canes are what will produce fruit for you next year, so don't cut them out. The canes producing this summer will die after fruiting and you will cut these dead canes out after fruiting/fall/next spring.

If you plants are primocane bearing, meaning that they have a summer crop and a second crop in the fall, then the new canes will fruit in the fall, then fruit again the next summer before dying.

OOPS! I went to look at the tag and see if they "primocane" and they are not raspberries at all! They are boysenberries! Is that different?

Related but different. You would manage them very similar to raspberries--they will fruit on last year's canes. I can't be more specific as "they don't grow here" so I've never had the opportunity to try them out. Somebody in boysenberry country needs to chime in with details.