Most commands have a single input channel (standard input, file descriptor 0) and a single output channel (standard output, file descriptor 1) or else operate on several files which they open by themselves (so you pass them a file name). (That's in addition from standard error (fd 2), which usually filters up all the way to the user.) It is however sometimes convenient to have a command that acts as a filter from several sources or to several targets. For example, here's a simple script that separates the odd-numbered lines in a file from the even-numbered ones
while IFS= read -r line; do
printf '%s\n' "$line"
if IFS= read -r line; then printf '%s\n' "$line" >&3; fi
done >odd.txt 3>even.txt
Now suppose you want to apply a different filter to the odd-number lines and to the even-numbered lines (but not put them back together, that would be a different problem, not feasible from the shell in general). In the shell, you can only pipe a command's standard output to another command; to pipe another file descriptor, you need to redirect it to fd 1 first.
{ while … done | odd-filter >filtered-odd.txt; } 3>&1 | even-filter >filtered-even.txt
Another, simpler use case is filtering the error output of a command.
exec M>&N redirects a file descriptor to another one for the remainder of the script (or until another such command changes the file descriptors again). There is some overlap in functionality between exec M>&N and somecommand M>&N. The exec form is more powerful in that it does not have to be nested:
exec 8<&0 9>&1
exec >output12
command1
exec <input23
command2
exec >&9
command3
exec <&8
Other examples that may be of interest:
And for even more examples:
P.S. This is a surprising question coming from the author of the most upvoted post on the site that uses redirection through fd 3!