How do simple bird waterers work?

Surface tension is plausible : it implies that, at the level of the hole, the pressure in water will be less than 1 atm. Thus the pressure isolines will look like this: enter image description here

Quantitatively, curvature $c$ will be of the order $1/(1 $mm$)$ because the hole looks about 1 mm size in your video, which with pure water would lead to a pressure difference drop of the order of 70 Pa across the interface. Then the flat interface with atmospheric pressure needs to be $c\gamma/(\rho g)$ lower than the hole, which is about 7 mm with pure water. Because you probably have some impurities, this will be less in practice. As @pwf suggests, you can use surfactants (soap) to reduce it (but that won't be a good thing for the birds!)