Why is a beam reach the fastest point of sail on modern sailboats?

The fastest point of sail depends on the boat (both its hull shape and its sail plan), the wind strength, and the sea state. In general, a beam reach is not the fastest point of sail.

For instance, in very light wind some boats will go fastest on a close reach due to the increased apparent wind from going toward the wind. For boats that sail faster than the wind, the limiting factor for speed is how close they can sail to the apparent wind; when they are going faster than the wind speed the apparent wind is always forward of the beam. A broader angle to the true wind allows them to go faster before their sails are sheeted all the way in, so a broad reach is the fastest.

Check out the polar diagrams on this site:

https://76trombones.wordpress.com/2009/10/17/polar-diagrams-vmg/

90 degrees to the true wind is not usually the fastest.

As for why some angles are faster than others, that's a bit complicated and beyond the scope of this answer. Suffice it to say that as you sail closer to the wind the component of lift in the direction of travel is less, plus there's increased drag. So you slow down as you approach a close-hauled course. As you bear away to a run, the sails eventually become stalled and less efficient. Somewhere in between a close-hauled course and a run you'll find the angle of maximum speed, which is sometimes approximately 90 degrees to the true wind but not always.

BTW, I should add that the term "beam reach" doesn't have a precise definition. Some sources say a beam reach is 90 degrees to the true wind, others say its 90 degrees to the apparent wind. Most sources introduce the points of sail before the concept of apparent wind and elide over the difference entirely. For boats that only go a fraction of the wind speed (i.e. the vast majority of them) its not that important a distinction. In the answer above, I've used the true wind definition, but even if you choose the apparent wind definition a beam reach is not always fastest. Boats that go faster than the wind have their sails trimmed to close hauled regardless of their point of sail.

On a beam reach, the boom is out over the side of the boat, and lift from the wind is a vector directed toward the bow, in the direction of travel.

On a close haul, the boom is near the centerline of the boat, and lift from the wind is a vector directed toward the side of the boat, almost orthogonal to the direction of travel.

A close-hauled boat depends on water pressure against the leeward side of the keel to keep it moving forward, rather than being pushed sideways.

On a beam reach, there is less water pressure on the leeward side, and the boat is able to slice through the water with less resistance.

This is a simplification, as actual sailing angles vary, but the higher speed on a reach is due to wind lifting the boat in the direction of its travel rather than pulling it sideways against its direction of travel, as in a close haul.

Another way of saying this is that if you consider the boat a damp and slick orange pit held between your thumb and forefinger (between wind and water), squeezing the tapered point of the pit will shoot it out, but squeezing the flat sides of the pit will have little effect.

The fastest course is not exactly perpendicular to true wind, but slightly downwind. The optimal downwind offset is the apparent wind angle that the boat can sail (beta), which is small for efficient boats (15deg) and iceboats (<10deg).

The geometry is explained here: http://en.wikipedia.org/wiki/Sailing_faster_than_the_wind#Maximum_speed_course_sailing_angle