Use it to correct someone's statement you believe is wrong, showing a different reality. You're selling your bike? On the contrary, I just bought a new one.
Pro Tip 2/3
Not a Conjunction
'On the contrary' is not used like 'but' or 'yet'; it stands alone to oppose a previous statement. You may think cats are aloof. On the contrary, they're very affectionate.
Pro Tip 3/3
Emphasis on Opposition
It's a strong way to emphasize you completely disagree with the previous point. Some say technology is harmful. On the contrary, it often enhances our lives.