Why do ion traps need high voltage? Why do we require high (order of kV) voltage in ion trap?
 A: You usually want to build your ion trap as big as possible and with as high a secular frequency as possible: Making it big, possibly even with several Millimeter distance between ion and the trap's electrodes helps you bring laser beams to your ion. Whilst those can in principle be focussed to half a wavelength beam diameter, with optics preferably outside your vacuum chamber, ten to a hundred micrometer are more realistic, but that can already make traps smaller than with one millimeter distance between the trapped ion and electrodes problematic: There is some intensity further away. Although it decreases exponentially with the distance, if you compare it to fluorescence you might want to see from your ion, even the small amount of light relatively far away can be unwelcome.
You want the secular frequency high for several reasons: First it helps in reducing heating (a gain in kinetic energy) of your ion because higher frequencies tend to carry less thermal and technical noise (and having a larger trap helps alot, too). But sometimes more importantly, you may want to resolve the motional modes of your trap; for lasercooling into the ground state, this is even a requirement. Finally, you want your ion trap's ponderomotive potential to be deep. Typically, you want to trap an ion formed from atomic vapor that may require a high temperature to form, so if you allow a bit of a safety factor, you will want a trap deeper than an Electronvolt or so, ideally obviously much deeper. All things equal, a deep ponderomotive potential again means a combination of high secular frequency and large size.
Yet a high secular frequency also necessitates a high driving frequency. The higher it is above the secular frequency, the smaller is your micromotion and the ion's sensitivity to (usually unintentional but to some extent always present) nonlinearities of your ion trap, which can make it unstable (not trapping) when a resonance involving a sum, difference, or multiple frequency (formed via nonlinearity) of the driving and the three secular frequencies (along the principal axes of the ponderomotive potential) occurs. Whilst understood in principle, as a researcher having built a trap, I feel at liberty to say that an experimentalist can be tempted into treating this aspect as a bit of black magic and introduce a bit of a safety factor here to reduce the risk of encountering problems of this nature.
All of these requirements are at odds with each other; actual ion traps are compromises. Not all of them operate above one Kilovolt, especially not those with only 100 micrometer or less distance between ion and electrodes, but such high voltages as you mentioned are indeed not uncommon. Unless you want to make your trap a poor compromise, that simply tends to be required. Note that there are many applications for ion traps and what I detail here may not include the driving factors for whatever use you are thinking of or are accustomed to expect.
