The solid angle is the extension of the concept of angle from two to three dimension. So let's start from 2d: consider a circle and pick two rays starting from the center. They will divide the circumference in two parts, called arcs. The length of each arc divided by the length of the radius will be the measure of the angle subtended by the arc itself.

Extend this to three dimensions: instead of a circle take a sphere, and instead of picking two rays pick a cone centered in the center of the sphere.The cone will cross the surface of the sphere: and now to define the solid angle measure the area of the surface delimited by the cone, divided by the square of the length of the radius (so that we have an area divided by an area).

The key point is that - since they are ratios - angles (and the solid ones make no exception) are dimensionless quantities: a small object as seen from a short distance can cover the same angle as a large object as seen from a long distance.

Why does this matter ? Because we live in 3 spatial dimensions ( :-) ). For instance consider a single light point source radiating (a star seen from very far?) By symmetry there is no reason for it to radiate more in one direction than in the other. So all the photons will be equally spread out in the space. Now you decide to look at how much light arrives in a given region of space: trace a "cone" from the region of space of your interest (the subject of your photo) with the vertex on the star, and you will have "measured" the solid angle. Now the ratio of photons will be equal to the ratio of the solid angle to the total (which is, by the way, 4*pi, similar to 2*pi in two dimensions): if the star is very far, this will be a very small number.

Now from stars move to flash units. These are not really point like (neither stars are, after all :) ) and not radiate isotropically (they are usually oriented so that all the light goes somewhere useful) but the same reasoning applies since they are usually much smaller than the subjects we are photographing.

This kind of computations underlies the so called inverse square law effect (basically you are spreading a fixed amount of light in a given solid angle: the area of the sphere subtended by the same solid angle grows with the square of the distance from the source, and so if you double the distance the area will be squared).