Here's an intuitive way to understand what's going on.
Imagine that at each node there is a person with a piece of pie, and at each timestep they divide up their pie and give it out to their neighbors in different proportions. For example, someone might like one neighbor more than the others, and therefore give them a higher percentage of pie at each step. At the same time each person gives out their pie, they simultaneously receive pie their neighbors are giving. What is the steady-state distribution of pie among the people?
Well, if someone receives as much pie from each neighbor as they give to that neighbor, for all people, for all neighbors, then that would certainly qualify as a steady state. Under certain additional conditions (most importantly, ergodicity), this steady state exists and is unique.
Let $p_{ij}$ denote the fraction of pie that person $i$ gives to person $j$ at each step, and let $\pi_{i}(t)$ be the steady-state amount of pie that person $i$ has at time $t$. Then the amount of pie person $i$ gives to person $j$ at time $t$ is $p_{ij} \pi_i(t)$, and likewise the amount of pie given from person $j$ back to $i$ is $p_{ji} \pi_j(t)$. The steady state balance, wherein everyone recieves exactly as much pie from everyone else as they give, is then: $p_{ij} \pi_i = p_{ji} \pi_j$
Now in the above reasoning we assumed the transition probabilities $p_{ij}$ are known and the steady state distribution $\pi$ is unknown, but we can turn this around and apply it to the situation where $\pi$ is given and we want to find some $p_{ij}'s$ satisfying the above equation so that they are consistent with having $\pi$ as a steady state. This is the idea behind the Metropolis-Hastings algorithm, and MCMC more generally.