I'd guess it comes from assembler, where each instruction has an address which may be jumped to by another instruction.
Additionally, the first computers didn't have much memory, and storing a line number only takes two bytes (if done properly). Writing a label takes more memory, first in the location, where that label is defined, then in any jump command.
Finally in the good old days there weren't any fancy editors. The only "editor" was a simple command line interface, which treated everything starting with a number being part of a program and everything else as commands to be executed immediately. Most prominent example should be the Commodore 64.
Newer dialects of Basic don't have the need for line numbers any longer.
