Shape of exclusion plots of WIMP

tl;dr: At high masses sensitivity decreases linearly with number density. At low masses detectors fight their energy thresholds. Exposure and backgrounds scale the curve up and down.

Full story: All these "direct detection" experiments look for simple non-relativistic scattering of dark matter with the nuclei in their detectors. Therefore, the same principles apply to all of them, and the curves all look somewhat similar.

All that matters in scattering is the reduced mass of the nucleus-WIMP pair. Once the WIMP is much heavier than the nucleus, the reduced mass doesn't change anymore, and so nothing changes in the kinematics or the experimental signature. Now, the mass density of dark matter is fixed, from the observation of the rotation curve of the Milky Way, to a mass of about a third of a proton-mass per cubic centimeter ($0.3 GeV/c^2/cm^3$). Direct detection experiments set limits on the interaction rate from dark matter, so fundamentally, they really just count the number of possibly dark matter-induced events that they observe in their detector during a given amount of time. Therefore, they really are sensitive to the flux of dark matter particles, or, to the number density of dark matter that we encounter on Earth. As you keep the mass density of dark matter fixed (to match observation), but increase the mass of your hypothetical dark matter particle, its number density decreases linearly with mass. Therefore, you expect to encounter less particles as their mass gets heavier. This means that all these curves have a simple 1/mass dependence at high WIMP masses.

Towards low WIMP masses, it becomes more and more difficult for the WIMP to kick the nucleus sufficiently to yield an observable signature. You know that from scattering: a fly can not transfer a significant amount of momentum when it hits a window. So at some point the experiments loose sensitivity to their energy threshold. Lower energy thresholds mean lower-mass WIMPs can still be detected and push that curve to the left.

In between these two effects there is a WIMP mass where the experiment's sensitivity is optimal. This is set mostly by the mass number of the detector target: heavier target nuclei (such as e.g. xenon) tend to have their optimum at higher masses than lighter nuclei (such as e.g. germanium) because it is harder for lighter WIMPs to transfer significant momentum to the heavy xenon nucleus. Also, experimental energy thresholds matter: the lower the threshold, the lower in WIMP mass is the optimum.

A perhaps counter-intuitive example comes from argon detectors, where you would expect a optimum at low masses (argon is light), however it is at rather high masses, because they have to fight detector backgrounds and therefore have a relatively poor (high) energy threshold that nullifies the effect they could otherwise exhibit due to their low target mass.

So much for left-right. Now, for up-down, the location of the minimum in cross section (in the vertical axis, a measure of the WIMP interaction probability) is set mostly by the "exposure" of the experiment and its background. The lower the background, the lower (=better) the minimum. Exposure is mass times observing time: the bigger the detector the lower the minimum, and the longer they waited for dark matter to show up, the lower the minimum. In the limit where there was no background in a detector, the vertical location of the exclusion curve would scale linearly with exposure.

An implicit assumption in these plots is that all dark matter is made from that one particle they look for. Therefore, in reality, the vertical axis isn't just cross section, but cross section times local density, which is set to $0.3GeV/c^2/cm^3$. If you just want a fraction of dark matter to be made from these particles, or think there's more or less of it around Earth for whatever reason, you can simply scale all these curves up and down.

I should mention one more effect: These curves assume that the interaction is "coherent", because the transferred momentum is so small that its corresponding wavelength tends to be longer than the size of the nucleus. In this case, the interaction rate (and thus location of those limit curves) scales quadratic in the mass number $A$ of the target nucleus. This is another reason why xenon experiments do so well.

All that said, these effects make for two main frontiers in the development of direct dark matter detection: One is the emphasis on "exposure" to push the limits to lower cross-sections, the other is the emphasis on "energy thresholds" to push sensitivity to lower WIMP masses.