Why do we have perspective in space? In Euclidean space the properties that we associate with perspective don't hold: parallel lines don't converge, there are no vanishing points, etc, but in vision we perceive space with perspective, i.e. as if it were perspectively projected.
How does our sight give rise to perspective? Is it a physical (as opposed to psychological) phenomenon?
E.g. when you have an object close to you it is perceived as big and far away as small.
 A: Assume you are looking with just one eye (in order to isolate perspective from stereo vision effects), and choose any Euclidean coordinate system that has your eye at the origin. The light emitted from any point on a given straight line will fall onto your retina at essentially the same position, i.e. straight lines through your eye are collapsed onto a single point. 
In projective geometry the geometry of projective space is studied. Points in projective $n$-space $\mathbb P^n(\mathbb R)$ are lines (1-dimensional subspaces) of an $n+1$ dimensional vector space. Lines are 2-dimensional subspaces, and so on. It can be coordinatized with homogeneous coordinates, which are $n+1$-tuples $[x_0:\cdots:x_n]$ describing the point in $\mathbb P^n(\mathbb R)$ represented by the line $\lambda(x_0,\ldots,x_n)$. Clearly we have $[x_0:\cdots:x_n] = [\lambda x_0:\cdots:\lambda x_n]$ for any $0\ne\lambda\in\mathbb R$.
Note that for perspective vision it might seem more natural to consider half-lines rather than lines, but since we don't have a more than 180 degree view that doesn't make a difference.
You can also study projective space using inhomogeneous coordinates, which are the coordinates in a plane that does not pass through the origin, most conveniently one of the $n+1$ planes where $x_i = 1$. This is essentially the projection onto that plane in the direction of your eye.

The inhomogeneous coordinates corresponding to $[x_0:\cdots:x_n]$ for $x_0 = 1$ are $(x_1/x_0,\ldots,x_n/x_0)$. This is a local coordinatization: every point of $\mathbb P^n(\mathbb R)$ can be expressed in some inhomogeneous coordinate patch, but not in all: directions parallel to the plane do not intersect with it, in the example those are the directions parallel to the $x_0 = 1$ plane.
You can imagine this plane to be your retina, or even an abstract plane of projection on which you project your image of the world, done very concretely by this renaissance artist on a woodcut by Dürer:

In this formulation it is easy to reason about perspective. Consider two parallel lines (parallel in $\mathbb R^3$), e.g. the sides of a straight road. In $\mathbb P^2(\mathbb R)$ they correspond to two planes that pass through your eye and these lines. These planes obviously intersect in a line through your eye. That line is (or rather projects onto) the vanishing point. 
As a final remark, note that projective space pops up in a non-directly-geometric context in quantum mechanics: there the state space is a complex Hilbert space of unit-norm elements, where two elements are considered equal when they differ by a scalar (which must have norm 1). This is exactly the same as saying that elements are 1-dimensional subspaces of the Hilbert space: nonzero elements that are considered equal when they differ by a nonzero scalar.
