Upon further viewing of the full time-lapse video, I can see your confusion. Watch the beginning. All of the legs have to twist to come out of the old shell. If they were solid it would be impossible. But since it's time-lapse, the final contortion after exiting the old shell is very fast. But in real life it would have taken many minutes. Once the crab is out of the shell fluid pressure within the body stretches out the legs. They have to be straight so that when the new shell forms they are already in the correct shape.

No they didn't 'instantly' solidify. But they will get a solid shell. You mistook time-lapse for actual time.

Sherlock.