Well, it’s very difficult to tumble in a circle. You need to build up a lot of speed to do double flips with a triple twist off the floor. If you run in a circle, you are constantly losing speed, and you won’t be able to do your skills.
Some people have a hard time, at least when younger, in doing their tumbling in a straight line. So it is important to learn how to stay straight.
The rules are that you have to use each corner of the mat. That’s why they do a few tumbling runs on one diagonal, and then work their way over to the other diagonal and do another tumbling run there. Judges are also looking for control, so in addition to the ability to stay in a line, they want to make sure you don’t step out of bounds. Stepping out of bounds indicates you’ve lost it—maybe your balance, or maybe just miscalculating your run to start the skills.