How can bats hear such high frequencies?
Short answer
Echolocating bats have relatively large sensory epithelia in their inner ear, that may correlate with their high upper frequency limit of up to 200 kHz. The basilar membrane is thinner and stiffer, possibly allowing it to decode higher frequencies. 

Background
In terms of the place theory of hearing, the cochlea acts as a frequency transformer, where high frequencies are encoded basally, and low frequencies apically. The cells responsive to sound, the hair cells, are located along the basilar membrane (BM). Going from base (high frequency) to apex (low frequency), the BM becomes wider and less stiff, making it more susceptible to encode low frequencies apically. Note that sound is encoded as a travelling wave, where the point of resonance of a particular frequency corresponds with the characteristic frequency of the hair cells (Fig. 2). 


Fig. 1. Rolled out cochlea. Source: New York University.

Fig. 2. Travelling wave and resonance frequencies along the basilar membrane. Source: University of Minnesota.

A study in Microchiroptera species revealed that the basal part of the basilar membrane was relatively narrow and stiff, which increases the resonance frequency of the basal part of the BM and hence may allow it to encode higher frequencies in these bats (Pye, 1966).

Because the high frequencies are encoded basally, it would be tempting to speculate the basal region would be enlarged in echo locating bats. Although this is true in some species, it may have more to do with frequency sensitivity than the upper limit of hearing, simply because a larger surface of the BM allows for more exquisite resolution of the BM. Although there is a tendency for larger BM lengths in echolocating bats, it is not consistent (Davies et al., 2013). 

The human BM is 33.5 mm, while the BM of the bat with the highest frequency limit known (200 kHz, Hipposideros bicolor) is only 9.13 mm. However, compared to that of acoustically unspecialized mammals of a similar body size, e.g. Sorex araneus (BM length 4.4 mm) and Mus musculus (BM 7.0 mm) it is relatively long. However, as said, other bats with lower upper-frequency limits have longer BM lengths, e.g., Pteronotus parnellii (13.2 mm) and Rhinolophus ferrumequinum (16.1 mm). Indeed, the BM length has been correlated both with an extension in frequency range as well as enhanced frequency resolution (Dannhof &amp; Bruns, 1991). 

References
- Dannhof &amp; Bruns, Hear Res (1991); 53: 253-68
- Davies et al., Front Zool (2013); 10: 2
- Pye, J Morph (1966), 118: 495-510