It's easy to understand why terminals used an 80 character line as that was the standard for the punch cards used in data entry. But why 24 lines? The reasons are less clear, but likely go back to the primitive technology of the 1960s and 1970s.
The technology in the 3270 was a generation more advanced than the 2260, replacing vacuum tubes and transistors with hybrid SLT modules, similar to integrated circuits. Instead of sonic delay lines, it used 480-bit MOS shift registers.27 The 40×12 model used one bank of shift registers to store 480 characters. In the larger model, four banks of shift registers (1920 characters) supported an 80×24 display. In other words, the 3270's storage was in 480-character blocks for compatibility with the 2260, and using four blocks resulted in the 80×24 display. (Unlike RAM chips, a shift register size didn't need to be a power of 2. While a RAM chip is arranged as a matrix, a shift register has a serpentine layout (below) and can be an arbitrary size.)
IBM provided extensive software support for the 3270 terminal.28 This had an important impact on the terminal market, since it forced other manufacturers to build compatible terminals if they wanted to compete. In particular, this made 3270-compatibility and the 80×24 display into a de facto standard. In 1977, IBM introduced the 3278, an improved 3270 terminal that supported 12, 24, 32, or 43 lines of data. It also added a status line, called the "operator information area". The new 32- and 43-line sizes didn't really catch on, but the status line became a common feature on competing terminals.Earlier terminals used a technology called sonic delay lines. I don't think that I'd ever heard of this until I read this article, and if you want to see just how primitive the memory technology of the days was, read this article.
Just for comparision with current tech, I'm writing this looking at a 28" monitor capable of 3840 x 2160 pixel resolution (although I have scaled it back to 1920 x 1080) displaying millions of colours. It's been less than 40 years since my monochrome 320 x 240 CGA display.
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