What does the one-tap equalizer used with OFDM accomplish?
The OFDM signal as a whole is affected by frequency selective filtering. It is usually designed such that the subcarrier bandwidth is smaller than the channel coherence bandwidth. This yields you frequency flat fading for each subcarrier which can be described by a single complex multiplication and equalization can equally done with just one tap.

Equalization in OFDM usually happens in the frequency domain. Let $X$ be the vector of subcarriers in the frequency domain and $x=IDFT(X)$ the corresponding time domain signal. Then, if the impulse response of your channel is a vector $h$, the received signal $y$ will be $y=x*h$ where $*$ referes to convolution. Then $Y=DFT(Y)=X \cdot H$ with $H=DFT(h)$ and $\cdot$ refering to element-wise multiplication. However, this is only true if $y$ is the circular convolution of $x$ and $h$ while in reality the convolution is linear. Therefore, the cyclic prefix is prepended (it needs to be at least as long as $h$) so that the linear convolution contains the circular convolution.

Then, $Y=DFT(Y)=X \cdot H$ with $H=DFT(h)$ and $\cdot$ as stated before. Since each element of $Y$ and $X$ refers to one subcarrier it can be seen that each subcarrier is multiplied with just one complex element from $H$ and hence equalization just becomes $Z=Y/H$ where again $/$ referes to element-wise division. This describes zero-forcing equalization which might not be the best choice if additive noise and interferers are present but it transports the idea.

And this is nothing else but one-tap equalization for each individual subcarrier.