Why can't water be liquid above 647 kelvins? In this article about vapor, it is said that liquid water can't exist if its temperature is higher than 647 kelvins. But if we put the liquid water in a closed container and heated up the water, couldn't it reach a higher temperature?
 A: In water, the critical point occurs at $647.096$ K ($373.946$ °C) and $22.064 \times 10^6$ Pa ($217.75$ atm). At this critical point, a second-order phase transition takes place. For pressures below the critical pressure, a first-order phase transition takes place by monotonic decreasing the temperature. The first-order coexistence (of water/vapor phase transition) line is terminated at that second-order critical point.
Interestingly, at temperatures/pressures higher than $647.096$ K ($373.946$ °C) and $22.064 \times 10^6$ Pa, normal water becomes supercritical water. In fact, above the critical point there exists a state of matter that is continuously connected with both the liquid and the gaseous state. The state of supercritical fluid can be found for any substance at a temperature and pressure above its critical point. The important property of this region is that the fluid can be transformed without phase transition into the liquid and the gaseous state, causing the vanishing of the latent heat. For these reasons, sometimes it is called as the fourth state of matter. You can put the liquid water in a closed container and heated up the water as much as you want and consequently it reaches a temperature higher than $647.096$ K ($373.946$ °C) but in such situation it is no longer a normal "liquid water" since there is no difference (distinction) between liquid and gaseous states of water. So, for the normal liquid water, the critical temperature (associated with temperature of the second order critical point) is the highest temperature at which liquid water can exist and this is the reason of that sentence (in that article in your post).
Please see the links in my answer for more details. Also see the phase diagram of water in the following figure from Wikipedia.

Warning: As noted in the comment by @Toffomat, the boundaries (green & red and orange & red are) of the supercritical fluid region in the $P-T$ phase diagram are misleading since there is no such sharp transition but instead a smooth change and thus the supercritical region is not as strictly defined as the above figure suggests. For this reason, the following figure (for ${\rm{CO}}_2$, but not water) seems to be more realistic in the supercritical region (from WikiPedia).

