This is a great question that I've thought of many times. The simple answer to "why" is because TV formats also drew their lines from left-to-right then top-to-bottom. Original computer monitors were CRT screens (small TVs), so the format naturally stayed the same. When monitors became flat screens (and TVs became flat screens too), it was equally natural to keep the same format for easy compatibility.
Of course you can then ask: why do TVs draw this way? They were invented in the early 20th century so you can imagine how much thought or lack thereof went into the design, if it was even questioned. Note: not meant to be disrespectful as it is incredibly more challenging to get equations for magnetically deflecting rays to the proper lines on a screen, as opposed to a simple matrix of tiny "light bulbs". (This kinda brings up the question of how on Earth did they invent CRTs before the simple matrix of dots, way back in the early 20th century?)
Nevertheless, my guess is that probably it was never questioned (a bad thing) since the Western languages write words left-to-right and top-to-bottom. Probably no one conceived of the possibility of doing it differently.
Personally, I dislike this format. I gained this attitude as I programmed games and other simulations involving elliptical orbits. Whenever you look up drawing equations involving sin, cos, or tan, you have to carefully invert the signs that deal with the y-axis parameters...or you will get the wrong thing drawn. An example would be the parametric equations for an ellipse drawn at any 2D angle. It can be a real nightmare dissecting the sinsin, sincos, and similar things.
Succinctly speaking in mathematical terms, the screen is in Quadrant 4 instead of Quadrant 1. This is unnecessarily complicated.
BTW when you get to 3 dimensions, the z axis is considered to go "upwards" in the positive direction. Kinda ironic. [EDIT]: Maybe not, see my comment below.
2 more things I realized/came across:
Sundials in the northern hemisphere (that have north-facing pointers and disks parallel to the ground) always rotate counterclockwiseclockwise. So if we put the zero hour at the top of the "clock face", then the shadow pointer starts moving to the right. This could be the origin of left-to-right direction in western languages, propagating down to Cartesian coordinates and TV/computer screens.
Old computers did not draw graphical objects so much. They drew text in a command prompt. So it is natural to put line 0 at y=0. If we had a bottom-left origin, the math of drawing textlines would be a bit more involved, and it could've been a big deal to those old slow computers (who even made a shortcut for incrementing by 1, for goodness sakes, called "++"). Plus you need to know the resolution of the screen, whereas if you just do line 0 at y=0, you don't need to know the resolution.