I'm not sure on the details of your particular blackberry variety, but here's the way it should work with any normal blackberry (the non-primocane fruiting kinds).

You get a blackberry root cutting, seedling, or small plant and it will begin to grow primocanes in the first year. During winter it will go dormant and then in spring it will re-emerge with foliage and flower growth on last year's primocane, which is now called a floricane. Additionally a new primocane(s) will begin to emerge. The floricanes will flower and fruit and then die after fruiting. The new primocane will continue to grow through the season and then go dormant in winter to start the whole cycle over again.

My guess is that the growth you have on your blackberry are primocanes or they may have not had enough chill hours.

Hope that helps!

These new branches that grew... if they came out of the ground then those are primocanes and won't bloom until next year. If they grew out of the existing canes, then they should grow to a short length and produce blossoms and fruit at their tips.
I'm speaking of the most likely scenario since we don't know what kind of blackberries these are. And some blackberries *do* grow berries on primocanes, but it's statistically unlikely you have those.