Let me share a simple proof I found during a childbirth class 8 years ago:
Let $x_1,\dots,x_d\in\mathbb{R}$ such that $1,x_1,...,x_d$
are linearly independent over $\mathbb{Q}$. Let $\epsilon>0$ and $a_1,\dots,a_d\in\mathbb{R}$ be
arbitrary. We want to show that there are $n\in\mathbb{Z}$ and $y_1,\dots,y_d\in\mathbb{Z}$ such that
$$|nx_i-y_i-a_i|<\epsilon,\qquad 1\leq i\leq d.$$
We proceed by induction on $d$, the case of $d=0$ being trivial. So let us assume that $d\geq 1$ and the statement holds for $d-1$ in place of $d$. The initial hypothesis is invariant under replacing $x_i$ by $nx_i-y_i$ for any
nonzero $n\in\mathbb{Z}$ and any $y_1,\dots,y_d\in\mathbb{Z}$, while the conclusion only becomes stronger. Hence by Dirichlet's theorem on simultaneous diophantine approximation we can assume from the beginning that
$$|x_i|<\epsilon,\qquad 1\leq i\leq d.$$
By the induction hypothesis applied to the $d-1$ numbers $x_1/x_d,\dots,x_{d-1}/x_d\in\mathbb{R}$,
there are $m\in\mathbb{Z}$ and $y_1,\dots,y_{d-1}\in\mathbb{Z}$ such that
such that $r:=(m+a_d)/x_d$ satisfies
$$|rx_i-y_i-a_i|<\epsilon/2,\qquad 1\leq i\leq d-1.$$
Moreover, this inequality also holds for $i=d$ if we set $y_d:=m$. Now let $n$ be the closest integer to $r$. Then
$$|nx_i-y_i-a_i|\leq |rx_i-y_i-a_i|+|(n-r)x_i|<\epsilon/2+\epsilon/2=\epsilon,\qquad 1\leq i\leq d.$$
The proof is complete.
Added on 24 November 2023. In fact the proof above is essentially the same as the one in Estermann: A proof of Kronecker's theorem by induction, J. London Math. Soc. 8 (1933), 18-20. It also appears in Section 23.8 of Hardy-Wright: An introduction to the theory of numbers.