The candle flame heats the air in the vase, and this hot air expands. Some of the expanding air escapes out from under the vase — you might see some bubbles. When the flame goes out, the air in the vase cools down and the cooler air contracts. The cooling air inside of the vase creates a vacuum. This imperfect vacuum is created due to the low pressure inside the vase and the high pressure outside of the vase. We know what you're thinking, the vacuum is sucking the water into the vase right? You have the right idea, but scientists try to avoid using the term "suck" when describing a vacuum. Instead, they explain it as gases exerting pressure from an area of high pressure to an area of low pressure.
A common misconception regarding this experiment is that the consumption of the oxygen inside of the bottle is also a factor in the water rising. Truth is, there is a possibility that there would be a small rise in the water from the flame burning up oxygen, but it is extremely minor compared to the expansion and contraction of the gases within the bottle. Simply put, the water would rise at a steady rate if the oxygen being consumed were the main contributing factor (rather than experiencing the rapid rise when the flame is extinguished).
