In this paper the authors refer to
transverse susceptibility $\chi_{ \perp}$ [meV $^{−1}$]
I was taught that the magnetic susceptibility is dimensionless.
How do I get $\chi$ in the above units??
Magnetic susceptibility is a static constant relating the amount of magnetization to the applied magnetic field; it is dimensionless. Transverse susceptibility is a measure of the magnetic response of a material to an alternating field; it is not dimensionless. The transverse susceptibility is often used when talking to the ferromagnetic resonance (like I think they are doing in the paper you cite) of a material.
So the short answer is that MS is dimensionless and TS is not.