If I fell at work and injured my back how should I properly get out of bed?

Carefully. You need to find a way with least amount of pain. Trial and error, see which way works best as you don't state which part of the back you injured. But, if the injury is that bed make sure to see a doctor.
Pain will tell you. "listen to your pain". Everyone's back pain is different. Your own pain is telling you what not to do. This will be more informative than any specific other instruction. As the only animal on earth that walks upright we get more back pain; fortunately, almost all such pain is time-limited. Usually avoidance of lifting while bending and twisting will help limit recurrences.