Why were arcade machines so big?

Because CRTs are big and heavy.

A picture tube of that era was generally about as deep as it was tall, so if you want to have a big picture, you're already going to have to have a cabinet that's fairly deep. Picture tubes of that era were also quite heavy. The 19" tube Pac Man used would have weighed in the 70 pound range. Bigger ones could be 100 pounds. They have to be mounted high enough for a standing person. You're not going to want to mount a very heavy CRT in some lightweight cabinet that could easily get tipped as it's going to be used by excitable teenagers yanking joysticks back and forth.

In addition to the size of CRT screens used in the 70s, 80s and 90s, the standard arcade cabinet was designed so that the average teenager could comfortably stand in front (sometimes so that two such teenagers could play side by side), easily access the controls and see the screen.

The machine also needed to have a fairly large bin for quarters (or local equivalent) and be reasonably accessible for owners and service techs to retrieve the quarters, or even perform minor repairs.

Arcade games also were not built using standard parts - some would use highly customised circuits and board layouts to suit the programme for the game. And, the computing power available in a modern smartphone was practically supercomputer level in the 1980s.

As you can see in this image of a Robotron: 2084 cabinet, the cabinets were far from empty: