--not 100% accurate, but a close approximation in lay terms--
If you want to measure the position of something precisely, you can bounce a laser off of it and time how long it takes the light to return.
If you want take this experiment to it's extreme in sensitivity, you eventually end up measuring tiny changes in the phase of the light bouncing off of the object. Your ability to measure position is limited by your ability to measure phase.
Quantum mechanics however requires there to be noise both in the amplitude and phase of the light. This is heisenberg's uncertainty principle. So measuring at the "quantum limit" means measuring fluctuations in phase equal to half of the uncertainty limit (half noise in amplitude and half in phase).
There is no law that said half the noise has to be in phase, just that the product of amplitude and phase noise is greater than or equal to the heisenberg limit. You can take extra noise out of phase and put it into amplitude to measure "beyond the quantum limit" without breaking any rules. This is called a "squeezed state."
That is exactly what they did. They created "squeezed light" and used it to measure a degree of freedom (like phase) "beyond the quantum limit"
This was a very challenging and impressive measurement. John Teufel has been doing work leading up to this for over 7 years (2 post doctorates and a professorship). He worked on quantum computers as an undergraduate too so you may even be able to add a few more years onto that.