A vacuum coffee maker operates as a siphon, where heating and cooling the lower vessel changes the vapor pressure of water in the lower, first pushing the water up into the upper vessel, then allowing the water to fall back down into the lower vessel.
Specifically, once the water in lower chamber is hot enough that its vapor pressure (the pressure exerted by the vapour component of a liquid) exceeds the pressure of a standard atmosphere, some of it begins to boil, turning into water vapor. Since the density of water vapor is about 1/2000 that of liquid water, the mixture of the air and water vapor in the lower chamber quickly expands, and, when the new pressure exceeds atmospheric pressure, pushes the remaining water up the siphon tube into the upper chamber, where it remains so long as the pressure difference between the upper and lower chambers is sufficient to support it (about 1.5 kPa or 0.015 atm).  This pressure difference is maintained during brewing through the continuous heating of the lower chamber. Coffee grinds are added to the water in the upper chamber.
When the coffee has finished brewing, the heat is removed and the pressure in the bottom vessel drops, so the combined force of gravity and atmospheric pressure overcomes the pressure of the bottom chamber, causing the brewed coffee to be pulled into the bottom chamber of the vacuum coffee maker, leaving the coffee grounds in the top chamber.
The iconic Moka pot  coffee maker functions on the same principle but the water is forced up from the bottom chamber through a third middle chamber containing the coffee grounds to the top chamber which has an air gap to prevent the brewed coffee from returning downwards. (Additionally, because the water is forced up through packed grounds, the pressures are greater.)  The prepared coffee is then poured off from the top.