projective plane is an ordinary plane equipped with a point at infinity.
For the real projective plane $\mathbb{RP}^2$ that would be a (projective) line at infinity, which itself is a normal real line plus an additional point, although that additional point isn't designated in any way, so calling it “the point at infinity” only makes sense in some cases, and none if you consider that whole line to lie at infinity.
With perspective, parallel lines only appear to be getting closer, they are not actually getting closer.
It depends. In space, they are not, but in the plane, they might be. Consider a pair of straight railway tracks on the ground. If you disregard the curvature of the ground and the thickness of the rails, this is a pair of lines in the plane. Now take a photo of that. On that photo, you will actually see these lines meeting, within the plane of the photo. This kind of effect is what a projective transformation can do. And the projective plane is the natural environment to express such projective transformations. For the projective plane, it makes no difference whether you talk about the ground or the photo: the lines will always meet. Simply because a projective transformation does not change the structure of the projective plane.
So, am I right that in projective plane we have two parallel lines that have constant distance of, say 10 cm everywhere, but at infinity they intersect? That seems crazy to me.
You are mixing worlds. Measuring distances is not a natural operation in the projective plane, and even if you define a distance measure which matches the euclidean one, it will not be defined for two points at infinity. Instead, the natural terms for projective planes are incidences, like which points lie on which lines, and perhaps projective invariants like cross ratios. In that setup, everything is sane. In particular, you get such beautiful facts like that every pair of distinct lines intersects in exactly one point, which now even holds for parallel lines.
