What is the relationship between posterior and full conditional probability?

They used a method called Gibbs Sampling, which is an instance of the more general Markov chain Monte Carlo method for computing the posterior distributions. The idea behind Gibbs sampling is that instead of trying to analytically compute the posterior distribution of lots of correlated variables (which is often very hard) you define a stochastic process so that, in the limit, samples from this process are samples from the true posterior distribution. The way this process is defined in gibbs sampling uses the fact that while sampling all those correlated variables at once can be very hard, computing the marginal posterior for a single variable is often easy. Then, if you can sample from the marginal distribution from each variable in turn, after many iterations, the expected fraction of time that variable has a given value is exactly the probability of that variable having that value, since you're effectively marginalizing the conditional distribution given many samples (I can point you to a proof that this converges if you want).

So their method of inference involves sampling each z_i using information from all the other variables in the model, which is why they use the equation P(z_i | z_-1,w).