$\def\HH{\mathbb{H}}$The following sketch of an argument is taken from Grothendieck's On the de Rham cohomology of algebraic varieties. You can also find a good discussion in Voisin's Hodge Theory book, volume 1, chapter 8. I talked about it a little on the last day of my Hodge theory class.
Note: Every step here is meant to be nontrivial.
Step 1: Using resolution of singularities, compactify $Y$ as $X \setminus \bigcup D_i$, where $\bigcup D_i$ is a simple normal crossings divisor and $X$ is smooth and projective.
Step 2: Let $M \Omega^{\bullet}$ be the complex of sheaves on $X$ obtained by pushing forward $\Omega^{\bullet}$ from $Y$. Argue that $\HH(X, M\Omega^{\bullet}) = \HH(Y, \Omega^{\bullet})$.
Step 3: Let $\Omega^j(\log D)$ be the following subsheaf of $M \Omega^j$: Near a point $x \in X$, with local coordinates $(x_1, \ldots, x_n)$, and where $\bigcup D_i$ is locally cut out by $x_1 x_2 \cdots x_k=0$, a differential form is in $\Omega^j(\log D)$ if it is locally of the form:
$$\sum f_{i_1 \ldots i_r j_1 \ldots j_s} \bigwedge_{1 \leq i_1 < \cdots < i_r \leq k} \frac{d x_{i_a}}{x_{i_a}} \wedge \bigwedge_{k+1 \leq j_1 < \cdots < j_s \leq n} d x_{j_b}.$$
where the $f_{i_1 \ldots i_r j_1 \ldots j_s}$'s are in the local ring at $x$.
So $\Omega^{\bullet}(\log D)$ is a subcomplex of $M \Omega^{\bullet}$. Define $Z^j$ to be the quotient sheaf $M \Omega^j/\Omega^j(\log D)$.
Show that $Z^{\bullet}$ is acyclic. Deduce that $\HH(M \Omega^{\bullet}) = \HH(\Omega^{\bullet}(\log D))$.
Step 4: Write out the spectral sequence for hypercohomology for $\Omega^{\bullet}(\log D)$. On page $2$ (or is it $1$?), all of the terms look like $H^q(X, \Omega^p(\log D))$. Observe that these terms are all finite dimensional, because they are the cohomology of a coherent sheaf on a projective scheme. Deduce that $\HH(\Omega^{\bullet}(\log D))$ is finite dimensional.
