I will take a stab at it.
There are three ways to use symmetric polygons as a grid to regularize movement. This can be done with triangles, squares, and hexagons.
With triangles, you can move to an adjacent triangle in 6 directions (3 sides and 3 corners). However when you move across a corner, you will be covering more distance than when you move across a side. This is less than optimum in some opinions (see below).
With squares, you can move to an adjacent square in 8 directions (4 sides and 4 corners). Again, when you move across a corner, you will be covering more distance than when you move across a side. Again, this is less than optimum in some opinions (see below).
With hexes, you can move to an adjacent hex in 6 dirctions (6 sides). There are no additional hexes that are only adjacent at a corner. This creates a situation in which you will cover the same distance no matter which direction you go in.
Why keeping the distance covered is important:
In any situation where terrain is an obstacle to one's path toward an objective, the effect of this obstacle is reduced immensely when using triangles or squares as the regular movement.
Think about it this way. There is an impassable mountain range running north/south that is 7 squares wide and I need to get from the left side of it to the right side of it.
I have two units that I need to get around this mountain range. Both unit's are placed such that their north/south position is in the middle of the mountain range (.i.e equally distant from the north end of the mountains and the south end of the mountains). Unit A is one square to the west of the mountains. Unit B is three squares to the west of the mountains.
If we move these units to the point where they can go around the north end of the mountains, they will get there at the same time even though Unit B had much farther to go.
Unit A should have to travel 3 squares north (a distance of 3) to get to the point where he can 'step around' the mountains.
Unit B should have to travel 3 sqares north and three sqares east (a distance of just over 4 - remember Pythagoros) to get there.
With a square grid, they get there in the same length of time. With a hexagonal grid, they probably don't. The longer the distances in question, the more 'realistic' a hex grid will be.
Please note that with shooting at a distance (mentioned in this forum as a possibility for bombardment), this effect will be even more important. Squares could allow you to place a bombardment unit so that it could hit two cities, even though on a real map it would be too far away to hit either of them.
In short, a hexagonal grid reduces distortion of distances. This is important for maneuver and range. I think it is a wonderful addition to CIV V.