Communication should travel via the shortest path necessary to get the job done, not through the "chain of command." Any manager who attempts to enforce chain of command communication will soon find themself working elsewhere.
Musk was concerned about what so many companies face today -- poor communication that slows things down and keeps things from innovating.
To Musk, information must travel in any direction, between all levels, regardless of your rank or position. If something needs to get communicated in a way that will increase efficiency and productivity, to his point, it has to travel the shortest distance.
He also added in his email to his employees that "if, in order to get something done between departments, an individual contributor has to talk to their manager, who talks to a director, who talks to a VP, who talks to another VP, who talks to a director, who talks to a manager, who talks to someone doing the actual work, then super dumb things will happen. It must be OK for people to talk directly and just make the right thing happen."
So what's the starting point to ensure this is happening across your organization?
Freedom and autonomy
Autonomy is one of the fundamental elements of what intrinsically motivates human beings, which leads to better performance.
Knowledge workers must have the right data, insights, and tools to make high-quality, agile decisions on their own that benefit the team. This takes trust on the part of leadership by putting the power in the hands of their workers.
That's why, even in the most efficiency-driven organizations, leaders need to give teams freedom and autonomy in calm times, to do what they do best. They need to practice innovating and coordinating without the manager in the middle.
In the end, the strength in good leaders comes from pushing authority down for others to make decisions on their own.
Good leaders understand that they don't have all the answers and that every person has something to contribute. It's a shift from the command-and-control, "chain of command" communication patterns that Musk mentioned earlier, to one where leaders and managers are relinquishing control so that individual contributors closest to the action are being empowered to act on their own.