A basic answer is that "the monotonicity formula places constraints on the shape of a minimal surface" e.g., you cannot have a lot of area concentrated in a ball if then later there is a (relatively) small amount of area. This, along with the convex hull property, already tells you a lot about the possible shape of a minimal surface.
However, I think you are looking for a concrete application. Here is one sample application that is somewhat different from what I have described above:
Suppose that $\Sigma_i,\Sigma \subset B_2(0) \subset \mathbb{R}^3$ are smooth embedded minimal surfaces and for any $f \in C^0_c(B_2(0))$ it holds that
$$
\tag{*} \int_{\Sigma_i} f|_{\Sigma_i} \to \int_\Sigma f|_\Sigma
$$
as $i\to\infty$ (this is a very weak notion of convergence of surfaces). Then, if $x_i \in \Sigma_i\cap B_1(0)$ has $x_i\to x$ then $x\in\Sigma$.
Proof: Assume $x\not \in \Sigma$. Then, there is $B_{3\epsilon}(x)$ disjoint from $\Sigma$. Choose a bump function $f=1$ on $B_{2\epsilon}(x)$ and $f$ vanishing outside of $B_{3\epsilon}(x)$. Now, by (*) we have that
$$
\int_{\Sigma_i} f|_{\Sigma_i}\to 0.
$$
Thus, for $i$ large, $B_{\epsilon}(x_i)\subset B_{2\epsilon}(x)$, so we have arranged that
$$
|\Sigma_i\cap B_\epsilon(x_i)|\leq \int_{\Sigma_i} f|_{\Sigma_i} \to 0.
$$
On the other hand the monotonicity formula implies that
$$
\frac{|\Sigma_i\cap B_\epsilon(x_i)|}{\pi \epsilon^2} \geq \lim_{r\to0}\frac{|\Sigma_i\cap B_r(x_i)|}{\pi r^2} = 1
$$
since $\Sigma_i$ is smooth (and thus nearly flat on small scales). This is a contradiction.
Note that if the $\Sigma_i$ are not minimal (i.e., if they don't satisfy the monotonicity formula), it is easy to find a counterexample to the above result by taking a flat disk $\Sigma$ and forming $\Sigma_i$ by gluing on a lot of "tentacles" which all have small area (and thus small contribution to the integral as in (*)). Thus, this is a nontrivial result.
(Note that I have not attempted to prove the most general version of this result, if you want you might see Simon's GMT book or other sources on minimal surfaces.)
There are many related applications of the monotonicity formula. A simple yet powerful one consists of White's proof of the Allard regularity theorem. See Section 1.1 here https://annals.math.princeton.edu/wp-content/uploads/annals-v161-n3-p07.pdf or Theorem 7.8 here http://web.stanford.edu/~ochodosh/MinSurfNotes.pdf).
