I can identify two parts of how this may be confusing. The first is vapor pressure with respect to ice, and the second is supersaturation.
First, let's look over the equation for phase transitions. The saturation vapor pressure can be described by the Clausius Clapeyeron equation. Solving that equation explicitly (i.e. not using Bolton's formula used in the link), we find that $$e_s(T)=e_s(T_0)\exp(\frac{L}{R_v}[\frac{1}{T_0}-\frac{1}{T}])$$
where $e_s$ is the saturation vapor pressure, $T_0$ is a given temperature (usually 273 K), $L$ is the latent heat of transformation, and $T$ is temperature. Note that I said latent heat of transformation, so it could be from one phase to another. Thus we can make a phase space diagram shown below using $L_f$ (latent heat of fusion), $L_v$ (latent heat of vaporization), and $L_s$ (latent heat of sublimation, gas to solid). The values for these can be found here.

Where all three meet is the triple point, but is more commonly referred to as the freezing temperature of water. Notice that the below it is where the liquid does not exist. Instead the gas goes directly into solid (deposition). In reality, liquid water can exist below this temperature (supercooled water), but it will eventually turn into a gas or ice.
I made a graph (below), similar to the phase diagram above, but does not clip off the clausius clapeyeron curve. Notice that below 273 K, the saturation pressure of vapor w.r.t. (with respect to) ice is lower than water. This basically says that it will first turn into ice before it turns into water. Then, above 273 K, the vapor saturation curve wrt water is less than ice- meaning it is easier to turn into liquid water above 273 K than it is for ice.

Next is the concept of supersaturation. Supersaturation generically refers to a state that is above saturation. In this sense, supersaturation can be identified by having a relative humidity over 100%. While some websites may give a different definition, I think of relative humidity as being the ratio of water entering the liquid (or solid phase), to water becoming a gas. So supersaturation means that there is enough water vapor in the air, at that temperature, to form a cloud.
I should note, that the process of making a cloud can be a bit more convoluted than this answer. For that, I recommend looking at this answer where I discuss Kohler theory.
So the short answer is, there is sufficient water vapor in the air to form a cloud. But because it is below freezing, the cloud is formed via deposition, much like how frost is formed.