Understanding what it means to be ''of general type''

To start understanding this, it's probably best to start with some examples.

First, the conjecture says that if a curve has Zariski dense rational points, then it is genus zero or one. This is known (Faltings).

Second, the conjecture, plus the Enriques-Kodaira classification, says that if a surface has Zariski dense rational points, then it is a rational surface, a ruled surface, an abelian surface, a K3 surface, an Enriques surface, an elliptic surface, a hyperelliptic surface, or a blow-up of one of these. A general type surface is simply one that is not any of those.

I don't know how many of these are possible to visualize but each of these has a much more definite structure and set of key properties than the class of all general type surfaces.

You could also try to understand general type surfaces through some positive examples like high-degree hypersurfaces, covers of products of two higher genus curves, and hyperbolic surfaces.

$X$ is of general type iff it is birational to its canonical model $X^c={\rm Proj}(\oplus _{m\geq 0}H^0(mK_X))$. Here $X^c$ has canonical singularities and $mK_{X^c}$ is a very ample Cartier divisor for some $m>0$. Thus there is an embedding $f:X^c\to \mathbb P ^N=|mK_{X^c}|$ and $\omega _{X^c}^{\otimes m}=\mathcal O _{\mathbb P ^N}(1)|_{X^c}$ under this embedding. This is a very useful characterization.

One of the main consequences/results about varieties of general type is that (up to birational isomorphism) they have good moduli spaces. Let $v = K_{X^c}^{\dim X}$ be the canonical volume which is given by the top self intersection of the canonical divisor on the canonical model. In any fixed dimension, these volumes belong to a discrete set and for any fixed dimension and fixed volume, canonical models are parametrized by a quasi-projective variety (see https://arxiv.org/abs/1503.02952 for some state of the art results, details and references). Eg if $\dim X =1$, then $v=2g-2$ and the corresponding moduli space has dimension $3g-3$.

An important feature is that by the easy addition theorem, $X$ can not be covered by by varieties not of general type, in particular by rational curves or abelian varieties (which tend to have many rational points).