This experiment involves weak measurement. The standard dichotomy you've learned, "observe the state, and the wave collapses", doesn't cover everything you can do with quantum mechanics; you can also measure some statistical information about the state, and only partially collapse the wave. These are called "weak measurements", and what Heisenberg's uncertainty principle says for these is that the product of the noise in the measurement and the amount of disturbance of the quantum state is at least $\hbar/2$, where $\hbar$ is the reduced Planck's constant.
What is going on here is described fairly well in the abstract of a 2002 paper of Ozawa:
The Heisenberg uncertainty principle states that the product of the noise in a position measurement and the momentum disturbance caused by that measurement should be no less than the limit set by the reduced Planck’s constant, $\hbar/2$, as demonstrated by Heisenberg’s thought experiment using a gamma-ray microscope. Here I show that this common assumption is false: a universally valid trade-off relation between the noise and the disturbance has an additional correlation term, which is redundant when the intervention brought by the measurement is independent of the measured object, but which allows the noise-disturbance product much below the reduced Planck’s constant when the intervention is dependent.
Stating this in easier-to-understand terms, the proof of Heisenberg's uncertainty principle assumes that the disturbance in the measured system is independent of the state the system is in. If you set up an experiment where the disturbance depends on the state of the measured system (it isn't immediately clear how you can do this), then for some states the amount of disturbance can be lower than what Heisenberg's uncertainty principle says. Ozawa also gives a modified formula for the uncertainty principle which takes this into account.
The experiment described by the article is a demonstration of this. The experimental disturbance is shown to be lower than what Heisenberg's uncertainty formula yields, but is still larger than that given by Ozawa's modified formula. This experiment undoubtedly wasn't easy to do, but it agrees with the standard predictions of quantum mechanics.