How can modern Arcade machines have thousands of games while the originals could only hold one?

In addition to the technical considerations laid out well by the other answer, there are also the business and psychological ones.

After people became accustomed to the idea of videogames at all, the standard of one game per cabinet was simply the way things had always been — inertia. Once it became technologically possible to house more than one and have the user decide which to play, industry still took a long time to actually try it in any serious way. For a start, the established patterns of cabinet self-advertising is easy to understand; it had static art all over it that tried to sell itself to the passerby, as well as the game software's attract mode. With multiple games possible, how would you rejigger all that without confusing everyone and/or giving short shrift to your super whiz-bang game?

Beyond that, there are squishier reasons too. It was possible you'd cannibalize sales by letting sites present more games with fewer cabinets bought. And since the games would have to be be swappable, it would also be possible for locations to engage in secondary-market trades on just the individual titles, possibly further cutting into sales.

It wasn't really till Neo Geo starting in 1990 that the concept really took off, and even then, the limit was six games, although I only ever saw four-slot machines myself. Since each game was on a cartridge, the machine had to have one slot per, and only got one tiny segment of the static art space; with hardly any room for printed instructions, a lot of games had to incorporate an instruction sequence at the start of play.

Neo Geo games all had to use the same set of controls in the same layout, too. No one thinks of that as odd today, but back then arcade machines were very custom about these things, generally, and as home consoles began to eat the arcades' lunch, they doubled down on that being a differentiating factor, to a large degree.