How the Markov Chain Monte Carlo ensure the stationary distribution converge to the target distribution?

I am reading about the MCMC but now I got a lot of questions.
Firstly, It says we could construct a markov chain which satisfy the detailed balance:p(z)T(z,z')=p(z')T(z',z) and we could sample from the proposal distribution T after we converge to the stationary distribution p(z). But why shouldn't we sample from p(z) but from the T(z,z')?
Secondly, why after we constructed this markov chain, it will ensure converge to the target distribution we want but not some other stationary distribution? The book says it satisfy the aperiodic and ergodicity. How the two properties make sure the stationary distribution is the distribution we want?
I am getting stuck here for several days. Thank you for somebody helping me out.