It is about eliminating a certain subset of unpleasant emotions.
Some emotions are caused by purely physiological factors for example some forms of anxiety and depression. Buddhist awakening does not address those emotions in cases where they are caused by purely biochemical factors.
Some emotions are caused by thinking. These are what Buddhist awakening addresses. Unpleasant emotions caused by thoughts disappear when the thoughts disappear. They are illusions produced by the mind. By quieting the mind with meditation and observing the activity of the mind one learns the truth of this and learns to see through the illusions. This is really the main point. There is another side issue which gets a lot of attention because it sounds mystical. That is that when the mind is quiet one may notice that in the absence of much of the usual mental activity, the feeling of self may disappear too. The feeling of self is actually produced by mental chatter. This experience has the advantage of making it easier to let go of some types of emotions that are based on ego like pride or status or selfishness. If you don't feel like you have a self it is easier to let go of egotistical attachments and aversions.
But the patterns of thinking one engages in over a long period of time become habitual and ingrained, probably wired into the brain. It takes a lot of work to unlearn long practiced modes of thinking. So Buddhist awakening is divided into stages. That is why you often hear that after an enlightenment experience a person has to learn to integrate the insight into daily life. It means they recognized how their thinking is causing a problem but they can't just think differently, they have to learn to think differently and undo all the damage a lifetimes worth of delusions have done to their thought patterns.
So Buddhist awakening doesn't even rid you of unpleasant emotions it only helps understand how your thinking is causing them which allows you to begin to change your thinking - still a long and laborious process.
Many people have a misunderstanding of what Buddhist awakening is, They think an awakened person will be nice, will be ethical, will be moral, will be happy all the time etc etc. An awakened person is none of those things. Unfortunately almost no one has any interest in addressing this misunderstanding and it leads people to trust spiritual teachers in ways they should not be trusted.