There are at least two effects at work. They're related and connected, yet I'd keep them apart:
Bowed strings have naturally a pretty bright, edgy sound – in synthesizer terminology, it's actually similar (albeit more complex) to a sawtooth wave. These hard edges in the signal sound kind of raspy. (Particularly pronounced when beginning notes with a hard attack at the frog.)
However, they only happen at discrete points in time. If multiple violins play together, then the transients will never be really in sync, even if all violins are playing exactly in tune. The result of this is a waveform that has the same spectral composition, yet sounds much smoother than a single violin, because the transients don't add up and are “lost in the crowd”.
Note that this effect doesn't require there to actually be multiple violins at all: it can also be achieved if you only record a single violin with a close mic multiple times, and layer the recordings (doing that gives a result quite similar to 70's “string synths”: thick and smooth, yet somewhat unnatural and not really like a true string section).
If you hear a single violin, your ears can pretty well focus on it (the sharp transients help with this, but they aren't required). Your ear zoom in, if you will. This can give the kind of intimate closeness that's needed for a folksy fiddle tune, but it also brutally highlights any playing details that aren't so desirable.
If multiple violins play physically together, they will necessarily sound all a bit different: different instruments have different timbre, the players will be a few cents off in pitch, and they can't sit all on the same chair. Hence there is no single point in space to focus on: you get a wide ensemble effect. Reverb also factors into this.
I have in the past tried to also emulate this effect with single instruments. It's quite tricky; you can get an approximation when actually setting up mics in a large room as if recording an orchestra, but then skip from chair to chair for each individual voice. Ideally also switch between a couple of different instruments, but this is actually not that crucial.
You definitely need a bit of stereo spread to pull this off convincingly. Which is what's most lacking in the recording you linked to (that actually sounds a lot like good old mellotron strings).