This is an interesting question. I tried to look online and did not find much information. People are much more interested in the propulsion aspect of the kick.
In any case, this is probably not a comprehensive answer, but I see a few factors. Note that the hydrodynamics of swimming is highly non-linear and turbulent, so any "explanation" will necessarily take out a lot of details of how this works. Also, I don't know exactly what proportion of lift is due to each factor I mention below.
1 - Speed effect.
Flutter kick is known to make you swim faster. If your feet are lower than your head and you move forward in water without kicking (either using only arms or being pulled by a rope for example), your feet will be lifted, just due to the direction of drag force. The faster you go, the higher the feet in water. This would likely be true even if the flutter kick performed but was not producing any lift.
2 - Variable geometry of the kick.
The up and down parts of the flutter kick are not symmetrical. Notably, in a good flutter kick, the ankle is loose, so that the foot presents a large area perpendicular to its movement in the downward direction compared to the upward direction. Therefore, we can expect a larger force on the down phase of the kick than on the up phase of the kick. [EDIT] The movement of the ankle can be seen for example in this video https://www.youtube.com/watch?v=makJKOu-8ds. One clearly sees that the top of the foot is in line with the tibia when going down, but angled to about 90 degrees when going up. This effectively changes the length of the foot/lower leg combo.[/EDIT]
3 - Boundary condition between water and air.
Vertical force can be seen as created to conserve the vertical component of momentum imparted to water while kicking. In flutter kick, in the phase where the foot is close to the surface, there is very little water to push up, while there is a lot of water to push down. Since air is much less dense than water, the momentum imparted to the air above the swimmer is negligible, and the small quantity of water above the feet at the top of the kick will result in less momentum imparted to the water, and correspondingly to a smaller downward force.
[EDIT] The feet should not go up too much as any part of the swimmer above water does not contribute to propulsion and can only sink the legs of the swimmer. While this third effect is only there for a small portion of the kick (probably between a few centimeters down from the surface to the surface on the upkick), it still contributes to the asymmetry. Of note, the force that needs to be exerted on the legs to keep them horizontal is quite small as the density of the legs is already close to that of water. This can be seen by the small size of floats used by swimmers held between the legs when practicing arm only drills Volumes of only a few liters are common, which suggests that forces of only a few tens of Newtons are required.[/EDIT]
The importance of factors 2 and 3 could be determined experimentally by swimming with the ankle braced in a fixed position (for factor 2) and by doing a flutter kick deeper underwater to see how it works (for factor 3).
