What do we learn from the Wronskian in the theory of linear ODEs?

Here is a typical use in an undergraduate textbook: to prove that for distinct $\lambda_j$ the exponentials $e^{\lambda_jt}$ are linearly independent. It has some applications on the more advanced level, but you were asking about undergraduate textbooks. Also notice: uniqueness theorem, even for linear ODE is rarely proved in undergraduate textbooks, at least in the USA. So for linear equations with constant coefficients, the notion of Wronskian permits you to find $n$ linearly independent solutions without an appeal to the unproved uniqueness theorem. Same applies to the proof that cosines with distinct frequencies are linearly independent.

Another application. How to write a linear differential equation of order $n$ satisfied by $n$ given functions $f_1,\ldots,f_n$? Here is how: $$\left|\begin{array}{cccc}w&f_1&\ldots&f_n\\ w'&f_1^\prime&\ldots&f_n^\prime\\ \ldots&\ldots&\ldots&\ldots\\ w^{(n)}&f_1^{(n)}&\ldots&f_n^{(n)}\end{array}\right|=0.$$ Expanding with respect to the first column, we obtain that the Wronskian $W=W(f_1,\ldots,f_n)$ is the coefficient at $w^{(n)}$, in particular, if all $f_j$ are analytic then the singular points of the equation are the zeros of $W$.

The significance of the Wronskian is not limited to differential equations. Consider a finite-dimensional vector space $V$ consisting of functions. (For example, polynomials of degree at most $n$). Suppose we have a basis $f_1,\ldots,f_n$. How to expand a function $f\in V$ in this basis? Write $$f=c_1f_1+\ldots+c_nf_n,$$ differentiate $n-1$ times and solve the linear system with respect to $c_j$. The determinant of this system is the Wronskian. This was the original goal of Heine-Wronski when he invented it.

For less elementary applications, type "Wronski map" in the cell "Anywhere" or in the cell "Title" in Mathscinet search.

Quite an important use of the Wronskian arises in the spectral analysis of the Hill operator $$\frac{d^2}{dx^2}+q(x)$$ when $q$ is periodic. This is the search of Floquet exponents.

This is in the same spirit as Piyush Grover's comment. The determinant $\det(x_1(t),\ldots,x_n(t))$ definitely deserves a name (not only in the context of linear ODE's). In such a lecture students could (and ,in my opinion, should) learn the meaning of the divergence of a vector field $F$. Having learned Picard-Lindelöf they are ready to understand the flow $\phi(t,x)$ as he solution of the initial value Problem $\phi'(t,x)=F(\phi(t,x))$, $\phi'(0,x)=x$, and for a small cube $x+[0,r]^n$ you can throw the edges into the flow to get after a short time almost the parallelepiped with edges $\phi(t,x+re_j)-\phi(t,x)$ whose (oriented) volume compared to the volume of the cube is $$v(t,r)=\det[\phi(t,x+re_1)-\phi(t,x),\ldots,\phi(t,r+e_n)-\phi(t,x)]/r^n$$ If you take the derivative $\partial_t$ at $0$ and the limit $r\to 0$ you get the divergence of the vector field (no problem to take time dependent vector fields).

The desire to make this more precise also motivates the theorems about differentiability of the solutions of initial value problems with respect to the initial values. Then you can throw quite arbitrary small sets into the flow and compare the evolved (oriented) volume with the original one by calculating them with the $n$-dimensional substitution rule.