I've never used one, so can someone tell me how the Sinclair
ZX81 keyboard works?

Because I'm seeing like 5 options per key, and only one shift key.

Like the "S" key: It's Save, ARCCOS,. "S", LPRINT, and that graphical symbol. How do you select between those?

I'm not even sure how you type backspace on this. shift-0?

I have looked up the manual

this has made me more confused

but the answer seems to be: it's a combination of shift and being modal.
The computer knows when you'd be typing BASIC keywords, so pressing "P" gives you PRINT in that case, no shift required.
you can also do shift-9 to get to graphics mode, which'll let you type the graphics.

and there's some way to get inverse text, but I'm not sure how. it doesn't have lowercase.

@foone the model 15 had a similar modal shift -- press FIGS to start sending numbers and symbols, press LTRS to go back to alphabetics. Or send a newline or whitespace, depending on the options install in the *receiving* end.
@foone or the Russian version, press "1..." for numbers, "A..." for Roman letters, and "РУС" for Cyrillic. (I'm not sure how the fourth row worked on their teletype)