Understanding detailed balance (crossposted from stats)

Understanding detailed balance (crossposted from stats)

I guess I understand the equation of detailed balance, which states that
for transition probability $q$ and stationary distribution $\pi$, a Markov
Chain satisfies detailed balance if $$q(x|y)\pi(y)=q(y|x)\pi(x),$$
this makes more sense to me if I restate it as:
$$\frac{q(x|y)}{q(y|x)}= \frac{\pi(x)}{\pi(y)}. $$
Basically, the probability of transition from state $x$ to state $y$
should be proportional to their ratio of probability densities. My
question is, how does MCMC fulfilling detailed balance yield the
stationary distribution?