A witch is a person who acknowledges, appreciates, and works with energies which reside all around and within us.
Being a witch and being psychic are two different things. Most people are born with varying degrees of psychic abilities. Just like most people are born with the abilities to draw. Some people draw stick figures and some people draw amazing masterpieces. Your grandmother is talking about your psychic abilites,not about you being a witch.
Starting carrying a notebook with you. Whenever you have a thought or feeling about something (anything) write it down immediately. It doesn't matter how off the wall or crazy you think it might be, write it down. Once a week, sit down and go through your thoughts and ideas (aka predictions) and highlight the ones that came true.
It might take some time to get used to doing this, but soon enough you'll see that your gifts (and yes... they are gifts) are accurate. The more accurate you are, the more confident you'll feel!
Embrace this gift!