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Familiar readers should recognize (13) as the exponential part of the multivariate probability distribution where $A$ is positive-definite. In trying to derive the constant for the distribution that makes the integral $1$, the reading brings us to equations (14) and (15).

What is the Jacobian and how does it allow us to rewrite the equation as (15)?

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See Note the comment which tells you that often the determinant of the derivative is also called the Jacobian. It allows you to rewrite the equation due to the transformation (aka change of variables) formula.

(note, too, that $|.|$ is used to denote the determinant by some authors)

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