@gung has a great answer. I would add one example to explain the "initiation" in a real world example.
For better connection with real world examples, I would like to change the notation, where use $H$ to represent the hypothesis (the $A$ in your equation), and use $E$ to represent evidence. (the $B$ in your equation.)
So the formula is
$$P(H|E) = \frac{P(E|H)P(H)}{P(E)}$$
Note the same formula can be written as
$$P(H|E) \propto {P(E|H)P(H)}$$
where $\propto$ means proportional to and $P(E|H)$ is the likelihood and $P(H)$ is the prior. This equation means that the posterior will be larger, if the right side of the equation larger. And you can think about $P(E)$ is a normalization constant to make the number into probability (the reason I say it is a constant is because the evidence $E$ is already given.).
For a real world example, suppose we are doing some fraud detection on credit card transactions. Then the hypothesis would be $H \in \{0,1\}$ where represent the transaction is a normal or fraudulent. (I picked extreme imbalanced case to show the intuition).
From domain knowledge, we know most transactions would be normal, only very few are fraud. Let us assume an expert told us there are $1$ in $1000$ would be fraud. So we can say the prior is $P(H=1)=0.001$, and $P(H=0)=0.999$.
The ultimate goal is calculating $P(H|E)$ which means we want to know if a transaction is a fraud not not based on the evidence in addition to prior. If you look at the right side of the equation, we decompose it into likelihood and prior.
Where we already explained what is prior, here we explain what is likelihood. Suppose we have two types of evidence, $E\in\{0,1\}$ that represent, if we are seeing normal or strange geographical location of the transaction.
The likelihood $P(E=1|H=0)$ may be small, which means given a normal transaction, it is very unlikely the location is strange. On the other hand, $P(E=1|H=1)$ can be large.
Suppose, we observed $E=1$ we want to see if it is a fraud or not, we need to consider both prior and likelihood. Intuitively, from prior, we know there are very few fraud transactions, we would likely to be very conservative to make a fraud classification, unless the evidence is very strong. Therefore, the product between two will consider two factors at same time.