This is presumably because a (relative) set of distances to 3 different satellites will narrow your location down to two singularity points, on opposite sides of the earth.
Not quite, the two points don't have to be on opposite sides of the earth.
A signal from a single satellite means the device knows how far it is from that satellite and the position of the satellite. This alone puts you anywhere that is exactly x thousand kilometers from the satellite on a really big sphere (much bigger than the earth).
Only some of this huge sphere will intersect with the Earth (which itself is roughly spherical). The intersection of two spheres is either a single point where they just touch (very unlikely) or a circle. This circle of intersection doesn't have to be a great circle (i.e. one that passes antipodal points). Imagine two footballs pressed together, the circle of intersection isn't the same diameter as either of the footballs, as you press harder the circles of intersection will get larger.
3 spheres will intersect at two points, but these, again might not be on opposite sides of the Earth. They could be relatively close together (which is commonly what happens when you're talking about huge spheres with a radius of the geostationary orbit of satellites).
4 spheres, or more, will intersect at a single point (or near enough given the relative accuracies).
The Earth isn't used as a reference sphere as it isn't a perfect spheroid. GPS accuracy is usually down to 5m or so, whilst elevation (in the UK at least) can vary by over 1000m. So it takes at least 4 satellites to get a location fix.