What it's doing is stopping IR from going through the window and heating the room. The outside of the window instead absorbs it all and gets hot.

The reason that doesn't transfer all inside is first because can dissipate part of the heat on the outside surface. Second, the glass on the outside is separated from the glass on the inside. Which provides insulation.

Here's direct quote from the original published paper that acknowledged that and kinda makes it seem like the authors of OP's article hammed it up a bit:

"The device is tested in Phoenix for 37 h, and the chamber with the TRC can have a temperature up to 6.1 °C lower than that with the glass slide (Figure 4B and Figure S7). We note that both chambers have temperatures higher than ambient temperature because the transparency in the visible wavelength range of the TRC allows solar heating. In practice, TRC can work with other existing (e.g., air conditioning) or emerging cooling technologies (e.g., subambient radiative cooler) to reduce the overall cooling energy consumption of buildings."

It's not going down its simpley never getting therr in the first place

> Wouldn't a warmer temperature outside naturally lead to a warmer temperature inside given enough time?

sort of? It's a question of how much is "enough" time. Good insulation can keep a noticeable temperature difference without active cooling or heating for a very long time. In addition, temperatures outside don't stay at peak forever. For instance, if temperatures outside can hit peaks of 50°C but temperatures inside never go over 30°C that's huge win even if you'd like it a bit cooler still. With clever engineering you can reach much more extreme temperature differences using only passive cooling and insulation, but not without making sacrifices to aesthetics and on other fronts.