Let's go through the article's abstract (emphasis added by me):
Quantum fluctuations of the electromagnetic vacuum produce measurable physical effects such as Casimir forces and the Lamb shift.
They also impose an observable limit—known as the quantum backaction limi on the lowest temperatures that can be reached using conventional laser cooling techniques.
As laser cooling experiments continue to bring massive mechanical systems to unprecedentedly low temperatures, this seemingly fundamental limit is increasingly important in the laboratory.
Right, conventional laser cooling cannot get a system below a certain minimum temperature.
This is essentially because lasers (or microwave sources, or whatever coherent field generator you care about) output a so-called coherent state, which has a finite width in both of its quadratures.
This limit is called, later in the abstract, the "quantum backaction limit", as we'll see in just a moment.
Fortunately, vacuum fluctuations are not immutable and can be ‘squeezed’, reducing amplitude fluctuations at the expense of phase fluctuations.
Right, coherent states aren't the only possible states of the electromagnetic field (or any other harmonic oscillator)!
It is possible to generate so-called "squeezed states" where one of the quadratures is more narrow than the other.
These squeezed states do not violate the Heisenberg uncertainty relation: you get squeezing in one direction at the expense of broadening in the other.
This is directly related to what the authors refer to as reducing amplitude fluctuations at the expense of phase fluctuations.
I'm not getting in the details on this because it's out of bounds for what OP is asking.
Here we propose and experimentally demonstrate that squeezed light can be used to cool the motion of a macroscopic mechanical object below the quantum backaction limit.
Ok fine.
They get beyond the "quantum backaction limit" because they're not using a normal coherent state.
They use a squeezed state.
We first cool a microwave cavity optomechanical system using a coherent state of light to within 15 per cent of this limit.
We then cool the system to more than two decibels below the quantum backaction limit using a squeezed microwave field generated by a Josephson parametric amplifier.
Yep, as we just said, using a squeezed state lets you get past the limit you have with normal coherent states.
From heterodyne spectroscopy of the mechanical sidebands, we measure a minimum thermal occupancy of 0.19 ± 0.01 phonons.
With our technique, even low-frequency mechanical oscillators can in principle be cooled arbitrarily close to the motional ground state, enabling the exploration of quantum physics in larger, more massive systems.
Ok so there we clearly see that they did not get to absolute zero.
They still had about 20% of a phonon (one quantum unit of vibrational excitation) in their oscillator, whereas absolute zero would be zero phonons.
They say that in-principle you can used squeezed states to get to arbitrarily low temperatures (i.e. arbitrarily low phonons).
That may technically be true, but to get arbitrarly low temperature you need arbitrarily much squeezing, which is very, very hard to actually do in the lab.
They're making an "in-principle" statement where the conditions for the in-principle thing to be achieved are totally unrealistic, and what's more, it's not even known whether the theory accurately describes the physical system in the parameter range you would need to get, say, $10^{-10^6}$ phonons (see comments under Emilio's answer for more on this).
Statements like that are still useful, because they tell the reader that there's no known hard limit to how far you can go with the squeezed state technique, i.e. the limits are entirely practical.
This is important because other protocols could have in-principle limitations.
For example, I might have some protocol which cools an oscillator but cannot possibly get below 0.3 phonons because of something baked into the physics.
In that case, you know that if you need a phonon number lower than 0.3, don't even consider using my protocol.
