ReBCO tape is the specific high-temperature superconducting material they're using.
Another important material is FLiBe, which is a liquid that I think absorbs the energy from the fusion reactor. I don't really understand the properties that make it particularly well suited to the task, but I gather it's important.
According to the article, FLiBe has the same heat capacity of water, but a boiling point over 14x higher (1430 °C according to the article). Melting point is 359 °C, 3.5x higher. I will speculate that its basically used as a water coolant with the phase shifts shifted up and out. I bet the heat exchangers are exotic, too, having to operate at such high temps! In fact I'd expect to see a pretty sophisticated cascade of exchangers.
Higher temperatures in the coolant loop are normally desirable for efficiency. In a heat engine, the hotter the hot side, and cooler the cold side, the more energy you can extract after all.
I don't really see why it's important for a fusion reactor though, where efficiency isn't really a concern at this point.
Well, the energy output of the reactor is limited by the amount of energy you can get out, which is limited by how much coolant you can move through it and how much energy the coolant can absorb without boiling/exploding. The MIT SPARC/ARC reactor designs are physically rather small, so it's possible that heat exchange could be the limiting factor in power output of an individual reactor.