That looks for: beginning of line, a series of non-quotes, a double quote, captures a series of non-quotes, a double quote and anything else on the line, and replaces it by the captured material.
$ sed 's/^[^"]*"\([^"]*\)".*/\1/' <<'EOF'
> xyz... rsync: "/home/path/to/file": Permission denied (13) rsync:
> "/home/path/to/file1": Permission denied (13) rsync:
> "/home/path/to/file2": Permission denied (13) rsync:
> "/home/path/to/file3": Permission denied (13)
Test on RHEL 5 Linux with GNU
sed, but only using features that would have worked in 7th Edition UNIX™ version of
Incidentally, a slightly simpler way to do it is with two substitute commands; change everything up to and including the first double quote to an empty string (that's a sequence of zero or more non quotes followed by a double quote); change everything after what is now the first double quote to nothing:
sed 's/^[^"]*"//; s/".*//'
Efficiency-wise, there's unlikely to be a measurable difference in the performance. In terms of ease of maintenance, I suspect the latter is less taxing on the brain cells.