Initially, there is little context for why the author inserted that formula there, so it is challenging to figure out what its purpose is. The formula's equivalence is made possible by the 'chain rule' given 'conditional independence' of the attributes. Open link and see slide 20:
Probability, Conditional Probability & Bayes Rule
The expression on the right hand side and its compliment expression is used to determine the probability of each class (binomial in this case) c and not-c given X (i.e. given all four attributes) further on down the website's page, though I didn't find it obvious. Then the class with the higher attribute becomes the classification for the iteration of those four attributes.
The author inserts:
$P(c|X) = P(x_1|c)\cdot P(x_2|c)\cdot P(x_3|c)\cdot P(x_4|c)\cdot P(c)$
right after he introduces Bayes Theorem in his tutorial, to set us up for Example 2, further down the page, under the subsection called "Example2." The expression and its compliment are used to calculate and compare the probabilities of c ('Yes') and c-not ('No'), given an iteration of four attributes.
The goal is to find the higher probability between
$P(X|c)P(c)$ and $P(X|c^c)P(c^c)$
so we can select the class with higher probability as to classify the four given attributes.
You can see a similar use of this in a subsection call "Bayes Theorem" here:
A practical explanation of a Naive Bayes classifier