How to understand the KL divergence term in Variational Autoencoders

I have no question with the mathematical derivations, but I am struggling to understand the intuitions behind the KL divergence term KL(q(z|x), p(z)). Intuitively, this pushes the posterior distribution of q(z|x) for every data point x to the aggregated prior distribution p(z). If p(z)~N(0, I), we are essentially mapping the mean of q(z|x) for every x to the same point, which I cannot understand intuitively. I feel it makes more sense if the objective is to map p(z) to q(z) or p(z|x) to q(z|x), but the derivation did not arrive there. I know it may be a bit late to ask this question, but I get stuck here in understanding the VAE paper. Can someone help?