How to be more sensitive?

It sounds like you have depression and you should talk to a doctor about getting on antidepressants. If you have a chemical imbalance that does not allow your brain to produce the serotonin it needs to function, you end up feeling nothing at all. Contrary to what most people think, depression isn't always a deep sadness - more often it is the lack of feeling for anything. Don't be afraid to talk to your doctor or a therapist. If you're young, ask your parents; if you're in college, see if your campus has a health center you can go to.

Dude, I have the same problem with you. My IDGAF attitude has been with me for at least the recent 5 years. Then, I went to exchange in my first uni year which made me relieved because the atmosphere was very relaxing yet I was still quite numb. So my numbness has cost me a potential friend there whom I thought would have developed a very good rapport if it wasn't for my numbness.
Anyway, at first, when I was still oppressing my feelings so that I was numb, I used the rationality it gave me to find out what I really wanted and its obstacles. Then I talked to my friends about it and see how they could help. Also, I tried to care more about others' feelings so that I didn't appear to be emotionless. As long as you open up, you will finally find what's disturbing you and you will feel an urge to change it. Try doing active sports like boxing if you need to let steam out, or create your own fighting zone at home where you can hit whatever you want.