It's not really about nixing unhappy thoughts. A lot of people misunderstand that as the purpose of meditation. It's possible to get into bliss states through some form of concentration exercises, and among my friends, we call them bliss junkies.
Rather, it's the avoidance of unhappy emotions and thoughts that create distortions, leading to neurotic emotional and mental patterns. These emotions and thoughts will arise on their own regardless of how you try to suppress them (and often times, suppressing them will make them grow stronger).
The basic practice is very simple at it's core: what arises will arises, what passes will pass. So when it arises, you allow it to arise -- and remain aware. When it passes, you let it go -- and remain aware.
As things that have normally been suppressed start surfacing up, it's as if you are "sweating out" these toxic thoughts. It will get worse before it gets better.
It goes deeper than that, since detoxifying isn't the ultimate aim either. It's the direct experience that what you think of as "self" is really an illusion. It's only after that, that things start get really interesting.
Disclaimer: I'm not a Buddhist teacher. Just a practitioner for more than 10 years and have read some of the texts. For best results, talk to a real teacher.
The illusion of self is a big deal in Buddhism and it can be confusing and daunting at first, but the concept behind it is really not that difficult to grasp. (Putting it into practice is a different challenge!)
One analogy I like to use is that of an automobile. The individual parts - the engine, the wheels, the chassis, the steering wheel, the gears etc - are all necessary, but individually, none of them are the car themselves. No one would look at a seat and say "that's a nice car!" Yet, we do just that in our daily lives.
We experience anxiety and fear and say "that's ME!". We go through several bodily experiences as "I, me, mine". (Which is NOT the same as saying this body is not you. Think of the car analogy. The tears are happening to you, but the tears are NOT YOU.)
Therefore, in all Buddhist traditions, we are taught to view all experiences as fundamentally lacking any self. This body, this experience, this thought, this taste/smell/touch, nothing has the self. Those are all beautiful but empty experiences.
Hope this answered your "illusion of self" question at least partly.
Buhddha showed we are all the same, at a certain level. One big pot of consciousness, if you will. Being disciplined at presence allows for the insight of this truth. And it's repeatable by others thorough practice of certain meditaion techniques. The illusion becomes an ovious point of dissonance at that point, thus the illusion of 'self'.
Some of the leading neuroscience is showing that what you conceive as the sense of self is something created by the brain. It does not exist the way you think it does.
That's not the illusion I was talking about though. It's one thing to read about that, but it's quite another to directly experience this. Like @kordless said, this is a repeatable process where you can see this for yourself. (And there's a great book written by a neuroscientist who also practices mindful meditation, and wrote about these two perspectives).
This is also not limited to the Buddhist tradition, though the sutrayana Buddhist teaches this as the primary thing and speaks the loudest about it. (There are ... realms, let's just say, beyond no-self and emptiness).
One-point concentration leads to very well-described states called jhana (dhayana). In deeper states, there can arise an orgasmic bliss, probably better than anything you can take.
There's a verse from one of Mitten & Deva Premal's songs that describe some of this bliss well:
I stood alone at the gateless gate
too drunk on love to hesitate
To the winds I cast my fate
and the remnants of my fear.
Concentration is not mindfulness meditation, though.
This stuff is hard to explain textually as it's more about feelings you have, but if I were to try.. during meditation bad memories will enter your focus and set off a cascade of negative reactions. But the deeper into the meditation you are the harder it is to shake your focus. Instead you end up observing these patterns of thoughts and how they evoke one another. Your let them have their moment in the sun and then they fade into the background as they failed to provoke you. Some stuff I've read suggests that the brain also may rewire these memories to permanently have less of an emotional impact. After many sessions you literally have to go out of your way to bring these memories back up!
