The name R2R, was originally written as R/2R, and refers to a resistor ladder network consisting of resistors of two values, R and 2R. Using only those values a ladder is constructed of resistors (either as discrete components or within a silicon chip). The ladder results in a voltage output of a fraction of a reference voltage depending on the bit value of a digital input, which is all you need for digital to analog conversion.
This paper isn’t a Noddy explanation, but it sets out things slightly more approachably than Wikipedia.
Most modern DACs use a delta sigma design rather than an R/2R implementation. This Wikipedia article describes the different DAC architectures under the Type heading.
An R/2R DAC can be implemented in silicon (TI make some) but there are a number of discrete component (usually surface mounted) designs. Some of them replicate the ladder in order to eliminate non-linear distortions. I think one output is subtracted from another and the difference inverted and applied to cancel such difference out of the final output. This may not be feasible in silicon, but I’m not sure.
As described the R/2R converts a multi-bit PCM input, but it is possible to do something similar for a DSD input. I’m listening to one now, the Holo Audio Spring. It has 8 resistor networks. 2 channels/PCMx2/DSDx2 where the second network is used to correct linearity as above.
As to why have they become popular, I think some people never bought into Sigma Delta. Certainly the Schitt guys converted a lot of folks back to R/2R with the Yggdrasil.
Edit: the R/2R architecture advantage over a switched resistor binary weighted DAC is that by using only two resistor values greater precision is possible than by attempting to match varying resistor values.