tl;dr they've used "scrambled" twice for two different things, but the laser beam after leaving Earth's atmosphere will be coherent and in no way "scrambled". They will use the laser to propel the craft but probably the craft will passively modulate the reflected laser beam back to Earth for communications as well. (think wiggling a mirror to modulate light from your friend's flashlight in order to send signals back to them.) How are the tiny spacecraft proposed by Project Starshot supposed to communicate with Earth?
You're in luck I think! I believe this is describing the open access 2021 paper Photonic solution to phase sensing and control for light-based interstellar propulsion by C. P. Bandutunga et al.
Reaching another star system within a human timescale requires traveling at a sizeable fraction of the speed of light. To achieve this, the Breakthrough Starshot program proposes a ground-based laser array to accelerate a gram-scale spacecraft through photon radiation pressure, requiring of order 100 GW of continuous wave optical power. The coherent combination of many lasers is one avenue to achieve a high optical power output. Here we present a photonic solution for optical phase sensing and control to enable the coherent combination of order 108 individual lasers, including the ability to sense and compensate for atmospheric distortions, assuming the laser system is ground based.
Monitoring a hundred million lasers and their phase errors
The primary problem is to keep one hundred million separate lasers optically coherent with each other. There are N(N-1)/2 or ~5x1015 possible pairs of lasers, so you would want instead to use some hierarchical approach, and that's what's described in Section 2. B. Hierarchical Locking Scheme in the paper.
This use of the word "scrambling" is first discussed in Section 1, introduction. It's basically a signal processing "trick" or technique for phase feedback in the control network using optical fibers, and has nothing to do with the laser light leaving Earth. Those beams will not be "scrambled" (but they will be modified to undo effects of the atmosphere, see next section).
The system employs digitally enhanced heterodyne interferometry (DEHeI) , using pseudo-random binary codes to uniquely identify and demodulate the interference signal from multiple lasers. This capability to uniquely identify optical paths permits the measurement of the optical phase acquired along the non-common light paths to each emitter in the OPA. Our architecture proposes a hybrid method that combines DEHeI multiplexed phase sensing with wavelength division multiplexing (WDM) to enable a multilayer hierarchical control scheme capable of individual phase control of an arbitrary number of optical emitters. This hybrid multiplexing platform is able to surpass the multiplexed sensing limits of both DEHeI and WDM, making the scale required for the photon engine attainable.
 D. A. Shaddock, Digitally enhanced heterodyne interferometry Opt. Lett. 32, 3355–3357 (2007).
Pseudo-random codes (Gold codes, actually) are also what GPS satellites and CDMA cell phones use so that they can all transmit on the same band of frequencies at the same time and yet our GPS chips and cell towers can hear each satellite or phone separately, and pick up its timing or voice and data information. In the case of GPS it's done with correlators. See for example 66 GPS channels for 22 satellites - why the factor of 3? and answers and links therein.
The other use of the word "scrambling" is unrelated to that
Instead of unscrambling the effect that the atmosphere has on light coming in, you're preemptively scrambling the light going out so that the atmosphere unscrambles it.
This refers to adaptive optics which large optical telescopes and even radio telescope arrays use to carefully compensate for wavefront errors produced by various atmospheric effects (it's nonuniform water vapor and electron density in the ionosphere for radio; Would Adaptive Optics be Useful in Radio Astronomy?, and convection turbulence for optical; astronomical seeing). All they are saying in this quote is that they will measure the distortion of light from an artificial guide star, in this case a laser in a satellite in Earth orbit, shining down and then pre-distort the 108 lasers to correct for it, so that the wavefront will be flat after passing through the atmosphere.
Basically there are two feedback loops here
The internal loop is all the phase sensors implemented using the fancy optics and fiber optic cables to make sure the lasers would all be mutually coherent, basically "singing the same song" if there was no atmosphere.
The external loop is the monitoring of incoming phase from the artificial guide star at each laser site, and that error is injected into the first loop so that a perfectly pre-distorted wavefront leaves Earth's surface and gets un-distorted by the atmosphere.