It is a matter of:
We wear it on our less skillful arm.
It makes sense for right-handed people to wear it on the left wrist because it is easier and gives more precision to manage it (set time, alarm, etc.) with the right hand while wearing it, and also more comfortable to leave the right hand free for anything else (and I guess to avoid damaging the watch too).
On a side note I remember when I was a kid wearing stickers preferably in the left hand so they got less damaged and could last longer (I am right-handed).
Edit: @AndrewMartin gives a link in the comments to left-handed watches.
Edit 2: I found this very interesting:
Wristwatches with analog displays generally have a small knob, called
the crown, that can be used to adjust the time and, in mechanical
watches, wind the spring. Almost always, the crown is located on the
right-hand side of the watch. This makes it inconvenient to use if the
watch is being worn on the right wrist. Usually, therefore, watches
are worn on the left wrist, even if the wearer is left-handed.
In exceptional cases, the crown is on the left side of the watch. This
is, for example, to prevent it from digging into the wrists of golf
Digital watches generally have push-buttons that can be used to make
adjustments. These are usually equally easy to use on either wrist.