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 (the atmosphere) to an area of low pressure (inside the bottle).

A page from Harvard goes into more detail about other factors. Here is a summary

Argument:
Oxygen is replaced by Carbon dioxide. So, there is the same amount of gas added than taken away. Therefore, heat alone most be responsible for the water level change.

Source of the Error:
A simplified and wrong chemical equation is used, which does not take into account the quantitative changes. The chemical equation has to be balanced correctly. It is not true that each oxygen molecule is replaced by one carbon dioxide molecule during the burning process; two oxygen molecules result in one carbon dioxide molecule and two water molecules (which condense). Remember oxygen is present in the air as a diatomic molecule. [A reader clarifies the water condensation in an email to me as follows: If the experiment were done with the sealing fluid able to support a temperature greater than 212 F and the whole system held above this temperature then the water product of combustion would remain gaseous and the pressure within the vessel would increase as a result of three gaseous molecules for every two prior to combustion and the sealing fluid would be pushed out.]

Argument:
Carbon dioxide is absorbed by the water. Thats why the oxygen depletion has an effect.

Source of the Error:
This idea is triggered from the fact that water can be carbonized or that the oceans absorb much of the carbon dioxide in the air. But carbon dioxide is not absorbed so fast by water. The air would have to go through the water and pressure would need to be applied so that the carbon dioxide is absorbed during the short time span of the experiment.

Argument:
The experiment can be explained by physics alone. During the heating stage, air escapes. Afterwards, the air volume decreases and pulls the water up.

Source of the Error:
The argument could work, if indeed the heating of the air would produce enough pressure that some air could leave. In that case, some air would be lost through the water. But one can observe that the water level stays up even if everything has gone back to normal temperature (say 10 minutes). No bubbles can be seen.

Argument:
It can not be that the oxygen depletion is responsible for the water raising, because the water does not rise immediately. The water rises only after the candle dims. If gas would be going away, this would lead to a steady rise of the water level, not the rapid rise at the end, when the candle goes out.

Source of the Error:
It is not "only" the oxygen depletion which matters. There are two effects which matter: the chemical process of the burning as well as a physical process from the temperature change. These effects cancel each other initially. Since these effect hide each other partially, they are more difficult to detect.