Metropolis-Hastings Algorithm
Assume the Markov chain is in some state $X_{n} = i$. Let $\textbf{H}$ be the transition matrix for any irreducible Markov chain on the state space. We generate $X_{n+1}$ via the following algorithm:
(a) Choose a proposal state $j$ according to the probability distribution in the $i$-th row of $\textbf{H}$.
(b) Compute the acceptance probability $\alpha_{ij} = \min\left\{1,\displaystyle\frac{\pi_{j}H_{ji}}{\pi_{i}H_{ij}}\right\}$
(c) Generate a uniform random number $U\sim\text{Uniform}(0,1)$. If $U<\alpha_{ij}$, accept the move and set $X_{n+1} = j$. Otherwise, reject the move and keep $X_{n+1} = X_{n}$.
MY DOUBTS
I know that Markov chains that are aperiodic and irreducible admit a stationary distribution. I also know that any distribution $\pi$ satisfying the reversibility condition is an equilibrium distribution.
As far as I have understood, the Metropolis-Hastings algorithm proposes a Markov chain whose stationary distribution is $\pi$ (the target distribution we are interested in sampling) and then simulate the Markov chain.
However, I am a little bit lost as to the interpretation of the algorithm itself. Could someone interpret it step by step to me?
I know such question is quite naive, but I am not able to understand it properly. Any help is appreciated.