There's a huge literature on this problem, and depending on who you ask you will get different answers. I'll come at it from the geomorphology perspective.
The most uninformative explanation is that sand moves around by the action of water or air. But, how it does so is dependent upon the beach feature/aspect you're trying to describe. Air moves individual grains around by picking them up and carrying them on short trajectories, hops, and collisions with the dry sand bed. Depending on the wind direction, sand is either moved further down the beach, or into the water (or off the beach completely). Windy days are really when this happens.
Water is more complicated. The two dominant processes are alongshore currents and large storm events. Over long times, large areas, and from the birds eye view, shore parallel currents cause the beach to become wider or narrower. These currents are energized by waves and they carry sand along with them. When currents converge or diverge you have a net gain or loss of sand causing the beach to erode or accrete. For a net gain (accretion), imagine a highway traffic jam: the convergence of cars would be analogous to the a buildup of sand. So sand isn't really being brought in perpendicular to the coastline, instead it's being carried parallel to the beach in the shore parallel current and the traffic-jamming (or the opposite) of the current-carrying sand leads to an accumulation of sand (or net loss).
Storms move massive amounts of sand around, but are really punctuated events (compared to continuous alongshore currents). Waves associated with the storm do drive alongshore currents, and so are subject to the same traffic-jam dynamics mentioned before. But they also move sand around in other ways too. For instance, on barrier islands storms are responsible for the sediment 'over-wash' process, which looks like a tongue/blob of sand extending from the front to the back of the island.
Beach cusps are an interesting feature too (google 'self-organized beach cusps'), but note that the explanation for their formation is debated. Again, it's the water moving sand around, but this is not helpful for describing why a highly repetitious pattern occurs. Personally I believe the only way to explain the dynamics is through the complex-system perspective, which suggests that many of the features you see are 'self-organizing', in short meaning that the sum of the parts (individual grains of sand being moved by water and air) is not equal to the whole (large scale organization).
This all goes to say, we know that water and air move sand around at the scale of individual grains, but how these processes translate to larger scale movements is a difficult problem and is actively being researched. The short answer is, in order to understand how sand is moving around, you have to decide to some extent what feature or aspect of the beach you are interested in, what time scale, and what spatial scale.