If you are thinking of BASIC dialects of the 8-bit home microcomputers, then those computers did not have text editors (unless you bought some word processor application). There was no way to have the entire BASIC program source code "open in an editor", like you would have when programming today. Programmer wouldn't even think about the program as a source code file, or text, really.
Example problem
So, lets say you have a simple program without line numbers in your head:
FOR I=1 TO 42
PRINT I
NEXT I
You boot up your computer. You have a prompt "ready" or something like that, and cursor sitting in next line. This is much like today's REPL environments of different scripting languages, though not really as strictly line based, more like screen based. So not quite like REPLs of today, but close.
Now if you start entering the program, you might get error after first line, because BASIC interpreter tries to immediately execute (and forget) it, and it doesn't make sense without NEXT to end the loop. This is not text editor where you edit text, this is where you give commands to the computer!
Partial solution
So you need some way to say, this is program line, store it! You could have a special command or just a symbol telling that hey, this is program line, store it. Let's imagine this:
#FOR I=1 TO 42
#PRINT I
#NEXT I
Ok, now our imaginary BASIC interpreter stored the program and you can run it. But now you want to edit the PRINT line. How do you do it? You are not in a text editor, you can't just move cursor to the line and edit it. Or you want to add another line like LET COUNT=COUNT+1 in the loop. How do you indicate where the new line should be inserted?
Working solution
Line numbers solve this in a very easy, if rather klunky way. If you enter a program line with a number that already exists, the old line gets replaced. Now the screen-based REPL environment becomes useful, because you can just move cursor to program listing on screen, edit the line on screen and press ENTER to store it. This seems like you are editing the line, when in fact you are editing text on screen and then replacing the entire line with new one from the screen. Also, inserting new lines becomes easy if you leave unused numbers in between. To demonstrate:
10 FOR I=1 TO 42
20 PRINT I
30 NEXT I
After re-entering line 20 with changes, and adding new lines, it could be
5 COUNT=0
10 FOR I=1 TO 42
20 PRINT "Index", I
25 LET COUNT=COUNT+1
30 NEXT I
More problems we just solved
Another important function of line numbers in this environment is, you can LIST, that is print on screen, a range of lines by specifying the line numbers. Without line numbers, how would you specify what part of program you want to see, when you are not in editor where you can just move cursor and scroll through the text? And same applies to removing lines, with line numbers it's easy, just ERASE 25 or something like that.
And then there's the usual benefit of being able to use line numbers as a language construct, at least as a target of GOTO AND GOSUB commands. This could be replaced by labels, but using line numbers is much simpler to implement in BASIC interpreter, which still was a definite bonus in a typical 8-bit home computer of the '80s.
Also, from user experience perspective, this is surprisingly easy interface for editing the code. Just type a line starting with a number to insert new code. Use LIST to show lines, ERASE to remove them. To edit a line, list it on screen, edit text on screen, and re-enter line. One paragraph to describe this. Compare trying to describe use of old text editors like edlin of DOS, or ed or ex of Unix: you need one paragraph just to explain how the user can exit them, when started accidentally!
Conclusion
Other answers explain how line numbers came to be, I'm trying to cover here, why the line numbers survived as long as they did, how they kept solving a real-world problem: They offered a way to do the actual programming without a real editor, in a very simple way. Once proper, easy-to-use full-screen text editors became the mainstream way to edit code, both with hardware limitations disappearing and when inertia of people adapting new things was overcome, then line number based BASIC dialects quite quickly disappeared from use, because the core usage problem they solved was no longer an issue.