A bug in the code that removes the escape characters will read beyond the last character of a string if it ends with an unescaped backslash character. Under normal circumstances, this bug would be harmless since sudo has escaped all the backslashes in the command's arguments. However, due to a different bug, this time in the command line parsing code, it is possible to run sudoedit with either the -s or -i options, setting a flag that indicates shell mode is enabled. Because a command is not actually being run, sudo does not escape special characters. Finally, the code that decides whether to remove the escape characters did not check whether a command is actually being run, just that the shell flag is set. This inconsistency is what makes the bug exploitable.
For more information, see The Qualys advisory.