Nitinol is a metal made of nickel and titanium.

It has shape memory and superelasticity.

How does the shape memory work?

How can a change in crystal structure have an impact on the shape?

The difference in crystal structure is quantitatively important. When you observe the hexagonal planes or the alloy, there is only one way (i.e. all ways are equivalent) to stack a second plane on a first one, but for the third plane there are several locations where you can stack it, and they differ by one atomic distance approximately. The change from one crystal structure to an other can hence make big deformations, like one atomic distance per plane thickness which is as atom diameter too.

 

As I believe to understand it:

 

One crystal structure is a bit more stable than the other, but the small difference permits a deformation to slip the atomic planes and adopt a very different new shape: superelasticity. Heat anneals the crystal, whereby the atomic planes slip back to the more stable structure and macroscopic shape.

 

This is only one effect. Some materials can change the shape several times and reversibly over the temperature.