Joint probability density change of variables: Transform two random variables while preserving probability, using their densities and a magic number called the Jacobian. Boom! New probability density, new insights.
The theorem you provided is related to the joint probability density of random variables. It states that if you have two random variables, X and Y, and you know their joint probability density function, f(x, y), then you can find the joint probability density of any other two random variables, u(X) and v(Y), as long as u and v are invertible transformations
The formula you provided gives the joint probability density of u(X) and v(Y) in terms of f(x, y) and the Jacobian of the transformation, J(u, v). The Jacobian is a measure of how much the transformation stretches or shrinks different areas in the input space.
This theorem is useful for a variety of problems in probability and statistics. For example, it can be used to find the distribution of the sum or difference of two random variables, or to find the distribution of a function of a random variable.
In the specific case of the image you sent, the theorem is being used to find the joint probability density of Y = X_1 + X_2 and Z = X_1 - X_2, where X_1 and X_2 are two independent random variables with the same distribution. The theorem allows us to find the joint density of Y and Z even though we don't know the joint density of X_1 and X_2.