I know that the determinant of matrix describes a change of volume of objects in space. The exp of a Matrix yields another matrix, therefore the determinant of it should also describe some change in volume in space. In 2D, as seen in the link, is the blue area the set of all possible solutions of a given ordinary differential equation?
If so, is this the reason why det(exp(M)) = exp(tr(A))?
Credit: Visuals by 3Blue1Brown Illustration
The blue area is not the solution set. The solution set in that video is $Ae^{Mt}$ with $A$ an initial value constant matrix. The blue area simply depicts where $e^{Mt}$ sends the unit square at different time points $t$.
About the formula:
I believe this is Jacobi’s formula. The reason for it is as follows:
The determinant is the product of eigenvalues, and the trace is the sum of eigenvalues. It turns out that the eigenvalues of $e^M$ are $e$ to the eigenvalues of $M$, and so the eigenvalues of $e^{Mt}$ are $e^{\lambda t}$ for each eigenvalue of $M$. Therefore:
$$\operatorname{det}e^{Mt}=e^{\lambda_1t}\cdot e^{\lambda_2t}\cdot e^{\lambda_3t}\cdots=e^{t(\lambda_1+\lambda_2+\lambda_3+\cdots)}=e^{\operatorname{tr}M\cdot t}$$