Simply, It is because 3G and LTE have the same clock timing for multi technology handsets. an example.
In LTE for BW=5MHz, there is 300 subcarrier {with 10% guard band, 4.5MHz/15KHz=300}
But we know that in IFFT/FFT transformation, Nfft should be a power of 2 (to speed-up the FFT operation). 300 is not a power of 2 and the next power of two is 512.
Fs = Nfft x Δf (because Fs=1/Ts , Ts=Tsym/Nfft and Δf=1/Tsym=15kHz)
For BW=5MHz, Fs=512*15kHz=7.68MHz => Fs=2*3.84MHz (3.84MHz is chip rate in UMTS).
We could reach to our timing goals by sub-carrier spacing equal to 7.5KHz or 30KHz also, but 15KHz is an agreement base on multicarrier transmission challenges (ISI , Doppler effect, …).
Also for 5G, 15KHz is the baseline numerology as below:
