I don't think they're proposing building a compressor for use in refrigeration. They're proposing a solid-state cooler/heat pump based on the elastocaloric effect:
As the shape memory alloy block is strained, it gets hot. This heat would be extracted - I guess by blowing air through the device to somewhere outside the refrigerator?
The air flow is then stopped, and the device unloaded and allowed to return to its normal shape. This causes it to cool. This part of the cycle will presumably cool the refrigerator casing somehow...
I wasn't able to find any physical designs of elastocaloric refrigerators. However there is a book on the magnetocaloric effect, and chapter 11 shows some basic ideas for how a heat exchanger might work.
By the way, I think the novelty in the recent research is their material is more reliable - shape memory alloys have been known for a long time, but their recent formulation has a longer life before it begins to break down with repeated stretching and shrinking.