The reason for this is Allard's regularity theorem.
Roughly speaking Allard's theorem says that if near a point of the support of a stationary varifold the varifold has unit density and area close to that of the ball of the appropriate dimension (for a 2-varifold it would be area of a disk) then the support of the varifold is smooth at that point.
More precisely, there is an $\epsilon>0$ and $r_0>0$ (depending on the ambient geometry and dimension of the varifold). So that if $\Sigma$ is an stationary $m$-varifold in $M$ , a Riemannian manifold, and a point $p\in spt \Sigma$ satisfies
$$\mathcal{H}^m(B_r(p)\cap \Sigma) \leq \omega_m r^m(1+\epsilon)$$
for $r\leq r_0 $ then $spt \Sigma$ is smooth near $p$. Here $B_r(p)$ is the $r$-ball in $M$ and $\omega_m$ is the volume of the unit ball in $\mathbb{R}^m$.
Allard's proof is unfortunately quite technical -- a good reference is Leon Simon's (sadly) hard to find book "Lectures on Geometric Measure Theory". Luckily, for your purpose there is a simpler version with a very easy proof due to Brian White (see here for the paper).
Specifically, suppose you have instead of being a stationary varifold you know that $\Sigma$ is a smooth minimal surface. If $p$ is a point of $\Sigma$ so that $$\mathcal{H}^m(B_r(p)\cap \Sigma) \leq \omega_m r^m(1+\epsilon)$$
then one has
$$|A|(p)\leq r^{-1}$$
here $|A|$ is the norm of the second fundamental form. In other words you obtain a quantitive bound on curvature.
How does this relate to your question? Well you have that $\Sigma_k$ converge to $\Sigma$ as Radon measures. Let $p\in \Sigma$ this convergence implies that
$$\mathcal{H}^m(\Sigma_k\cap B_r(p))\to \mathcal{H}^m(\Sigma\cap B_r(p))$$
(it is worth noting that we are using that the surfaces are area minimizing to ensure the convergence is with multiplicity one).
Now since $\Sigma$ is smooth near $p$ it is locally modelled on a flat plane. In other words, there is a scale $r$ so that
$$\mathcal{H}^m(B_r(p)\cap \Sigma) \leq \omega_m r^m(1+\frac{1}{2}\epsilon)$$
hence by the convergence
$$\mathcal{H}^m(B_r(p)\cap \Sigma_k) \leq \omega_m r^m(1+\epsilon)$$
and so since the $\Sigma_k$ are smooth (either a priori or by Allard's full theorem)
and the point $p$ was not important we deduce that for some $\delta>0$ we have
$$\sup_{B_\delta(p)\cap\Sigma_k} |A|\leq r^{-1}.$$
In other words there is a uniform curvature bound on the $\Sigma_k$. The result then follows from "Standard elliptic PDE" and the Arzela-Ascoli theorem.
