Why do we need FFT pairs for OFDM?
The short answer is: Complex exponentials are eigenfunctions of an LTI system and the IFFT modulates different complex exponentials with the QAM data.

More elaborate: The wireless channel can (if it is time-invariant for the duration of one OFDM symbol) be modeled as an LTI system with impulse response $h[n]$ (in discrete-time). So, the received signal is the convolution of the input with $h[n]$. At the receiver, you want to know the data, so if you did directly transmit the data, you would need to perform a deconvolution at the receiver.

One can understand the FFT/IFFT to exploit the convolution theorem (convolution in time is multiplication in frequency). We transmit a signal in time domain at the transmitter, and transform it into the frequency domain at the receiver. Then we know, that the frequency response of the channel has multiplied the frequecy domain of the transmit signal. So, it becomes obvious that we directly define the frequency domain of the transmit signal with our QAM symbols. 

This is not the whole story (circular convolution with the impulse response must be achieved such that the convolution theorem holds in discrete, and we use a Cyclic prefix to obtain it). 

I've written some articles about OFDM, namely an OFDM walkthrough example and about the CP . Maybe these articles can help you understand better. I've always wanted to write something about the math behind OFDM, but didn't find the time yet.