The liquid xenon is serving a dual purpose.
First, the xenon nuclei serve as a target for the WIMPs to interact with. There are a number of hand-wavy reasons why heavy nuclei would be more likely to interact with most kinds of WIMPs than low-mass nuclei: more mass density, more charge on the nucleus, more electron density in case the WIMP interaction is with the electrons rather than the nucleus, and probably some other reasons. The details, and the theoretical models which justify them, occupy a chapter in most PhD dissertations on WIMP searches.
Second, the liquid xenon acts as a scintillator, transforming deposited energy into light which can be carried to a photodetector. This is true regardless of the source of the ionizing radiation: the xenon also scintillates in response to cosmic ray muons and intrinsic radioactivity.
Noble liquids are nice to work with experimentally because they are self-purifying: any chemical contaminant which is not xenon will freeze out of the liquid. All of the noble liquids have similar chemistries and behave as scintillators. (There are some neutron experiments which use liquid helium as the same kind of target-plus-detector combination.) But if increasing the nuclear mass increases the cross-section for interacting with WIMPs, you want your target to be made of the heaviest available nucleus. And the heaviest non-radioactive noble gas is xenon.