Skimming the paper, I believe the relevance of the wind stress curl is its relation to "Ekman pumping". I haven't found a simple, concise reference for this, but this page might be a good start, and this page has a couple of formulas about wind stress curl. I'll try to summarize here.
When wind blows over water, the top of the water starts moving. It shears against the water below it, so that water starts moving too. The momentum from the wind is transferred down into lower layers of the water. This water also feels the Coriolis force. The direction it ends up moving in depends on the balance of friction/drag and Coriolis force. On average, the water moves to the right of the wind in the northern hemisphere; if the wind is blowing northward, the water moves eastward.
Now imagine you have strong wind blowing northward at one location and weaker wind to the right of it. The water at the first location moves to the right, and it does so faster than the water at the second location (because the wind forcing the water is stronger at the first location). The water converges at the second location, pushing the water downward. This is how the curl of the wind stress (the northward wind changing in the east-west direction) is related to the water convergence (the eastward current changing in the east-west direction) and hence to water being pushed down or pulled up. Positive wind stress curl pulls water up; negative wind stress curl pushes it down.
The last relevant part here is that this kind of motion suppresses ocean mixing. The relevant sentence from that paper is
The wind stress curl and hence Ekman pumping anomalies were negative, which also is consistent with relatively weak entrainment.
"Entrainment" is how much of the deep, cold ocean water mixes with the relatively warm upper ocean water, cooling it. The negative wind stress curl leads to water being pushed down and less deep water mixing with the upper ocean. The upper ocean stayed warmer, so the whole heat blob lasted longer.
