Buying stocks priced under $20 is mostly about investor psychology. It's an extreme reflection of the mantra to buy low and sell high.
Math also plays a part. A 10% gain on a $20 stock means the stock only has to move $2. For some of these stocks, that can happen in a matter of days. A similar gain on a $200 stock requires a $20 move. All other factors being equal, the lower-priced stock will have a higher percentage volatility driven by factors such as a lower float or less institutional ownership.