They were used as labels for GOTO and GOSUB
Like this:
10 PRINT "HELLO WORLD"
20 GOTO 10
There were no named labels in some early BASIC versions
They were also required if you wanted to insert a line between 2 existing lines of code, because in the early days, you had no full text editors. Everything had to be typed in the "interactive" interpreter.
So if you typed:
15 PRINT "AND THE UNIVERSE"
The program would become:
10 PRINT "HELLO WORLD"
15 PRINT "AND THE UNIVERSE"
20 GOTO 10
When you ran out of line numbers, you could run a "renumbering" tool to renumber all lines in your program, but in the very early days of the Commodore 64 and other home computers, we didn't even have that, so you'd have to renumber manually. That's why you had to leave gaps of 10 or more in the line numbers, so you could easily add lines in between.
If you want to try out the Commodore 64 interpreter, check out this C64 emulator written in Flash: http://codeazur.com.br/stuff/fc64_final/ (no install required)