Probably this description is going to be the most unscientific and biased possible explanation, but it is just my experience.
I started meditating in 1982, my life was pretty much a disaster no future ahead, but I don’t want to focus on that. The point is that I decided to go through the initiation process, that I learned many years later that the short ceremony, taking flowers, fruits and listen to a “pusha” which is like a poem that the instructor recites, was not intended to me but to the teacher. It is a reminder that he is a just a channel administering the knowledge, he doesn’t own that knowledge hence he cannot modify it. He needs to observe the protocol and nothing will be changed; reason why it doesn’t matter where you learn the TM technique everywhere is the same.
The process was pretty simple and I started meditating in no time, it is an individual one and yes you can go to the TM center or go to retreat, usually on a weekend, but that is another story and certainly not a requirement.
At the beginning you just fell very relaxed after meditating for 20 minutes, and as you keep on practicing you go to deeper and deeper levels of interiorization, which is amazing, but don’t give it for granted because the experience is necessarily supported by the physiology. But what does it means, well if you are under stress, not sleeping well, sick, in general whatever affects your physiology your experience is affected, meditation still works but your starting point is way lower, so even though you feel better and relaxed after meditating you no longer reach that beautiful deep experience.