The short answer is that it wasn't - in the beginning, both directions were used, though perhaps right to left was used more often. Many early inscriptions are in what is called Boustrophedon, or ox-turning, where the text goes first one direction, then goes back, sort of like plowing a field.

As to why a person might prefer one direction to another, this is likely connected to the medium on which a person is writing more than anything else. Lets say you decide to start using writing to inscribe on your weapons. Are you inscribing the handle of your spear? Then both directions being available, you'll start from the natural divide - the split between the head and the shaft, and work down. If you're right handed, holding a spear in front of you, that means you'll read left to right. Is it a sword you're holding? the natural division is now between the handle and the blade, and writing up the blade as a righty is going to be right to left.

In the case of a sea-going people, the distinction between port and starboard (steering-board) is pretty ancient - if you always pull up to port on your left side, and want to have your boat named, starting from the front, it's going to read left to right. If you're a culture that is writing out it's sacred texts on vellum, and has a tradition of left-bad right-good, maybe you use skins from the right sides of sheep, starting at the head, and as a result go right to left.

As you can see, the reasons are endless, but generally are going to be connected to the medium being written on as the alphabet was becoming established in a given culture; very soon the main reason becomes tradition, and little inconveniences like smearing the ink have far less to do with it.

Care to clarify which writing systems you're talking about? Both directions are used today and have been used for thousands of years.