Multi-head attention just means that you're looking at all the words at once rather than only looking at one word at a time, and using that to generate the next word. So instead of using attention only on the last word you also have attention on the penultimate word and the one before that and the one before that, etc. I think it is fairly obvious why this gives better results than say an RNN – you are utilizing context better than in a recurrent system, which is also just closer to how a human brain works. When you read/write a sentence you're not really going one word at a time, you're thinking about all the words at once, even if the last word is technically the most important.

The other clear benefit of transformers over an arch like RNNs (and what has probably made more of a difference imo) is that its properly parallelizable, which means you can do huge training runs in a fraction of the time. RNNs might be able to get to a level of coherence that approaches GPT-3, but with current hardware that would be very time-prohibitive.