From the (Mac OS X, but others are similar) man page:
strip removes or modifies the symbol table attached to the output of
the assembler and link editor. This is useful to save space after a
program has been debugged and to limit dynamically bound symbols.
Note the bit about "after a program has been debugged" because debugging a stripped executable is very hard, indeed. The "limit dynamically bound symbols" is a rarer use: it lets you make some of the functions in an external library inaccessible by taking away the index entries that tell where the actual code is located. This is also explained in the man page.
As cheap and plentiful as disk is in most situation you simply wouldn't bother with this anymore. But you might want it for space constrained situation like embeded devices, rescue disks, etc.
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