(So, this is an unbelievably naïve attempt at understanding the conjecture; I literally just heard about it for the first time today.)
(Before I begin, I would like to give major props to David Metzler https://www.youtube.com/watch?v=gIi92JSZ9J4 and Aleph_0 https://www.youtube.com/watch?v=Jqbvat1fhPI for their wonderful videos explaining to me and introducing to me, respectively, The Hodge Conjecture.)
So, it is a theorem of Bézout that if algebraic varieties $V$ and $W$ have degrees $n$ and $m$ respectively, they have $nm$ intersection points.
But, as David Metzler explains it, there are three major impediments to finding/counting the intersections of algebraic varieties. 1) The varieties may not have any real intersections, or, at least, fewer than are predicted by Bézout's Theorem; for this, we use complex varieties. 2) The varieties, like parallel lines in the plane, may not intersect in the finite portion of $\mathbb{R}^N$; for this, we use a projective space. Hence, to consider intersections of varieties, we use $\mathbb{C}P^N$ (for $N$ large) as an embedding space for all the varieties in question. Note that by a confluence of about 10 theorems, any complex submanifold $M$ of $\mathbb{C}P^N$ is Kähler. Finally, 3) an intersection may be a "multiple" intersection, and so to correctly "count" the number of intersection points, the intersection points may need to be "counted with multiplicities"; to get around this third point, we assume all complex varieties are smooth manifolds and make both varieties transverse.
So, in $\mathbb{C}P^N$, we consider a smooth complex variety $V$ with $2n$ intersections with any (and, hence, all) lines $L$, adjust $V$ to a complex manifold $M$ with $M$ transverse to $L$, so they genuinely intersect in $2n$ points. Now, consider an $H^{2n}(M; \mathbb{Q})$ Poincaré dual $w$ to $M$. As $M$ is Kähler, $w$ has a unique representation as harmonic differential forms $\sum_{p+q=2n}r_{p,q}\omega_{p,q}$. By black magic, the integrals $\displaystyle \int_{V'} \omega_{p,q} = 0$ for $(p,q) \neq (n,n)$, so, for reason to which I'll come back some other day, we need only consider $\omega_{n,n}$.
The Hodge Conjecture asks if there is a collection of complex varieties $V_1, V_2, \ldots, V_m$ with $\omega_{n,n} = \sum_{i=1}^m q_i[V_i] \in H^{2n}(\mathbb{C}P^N; \mathbb{Q})$, where each $[V_i]$ is a rational Poincaré dual to $V_i$; that is, The Hodge Conjecture asks if $\omega_{n,n}$ must be algebraic.
I think this is the coolest open problem in manifold topology I have ever seen (with possible apologies to The Borel Conjecture).
