Bash script error with strings with paths that have spaces and wildcards

Inside quotes, the * will not expand to a list of files. To use such a wildcard successfully, it must be outside of quotes.

Even if the wildcard did expand, the expression "${FILES}" would result in a single string, not a list of files.

One approach that would work would be:

#!/bin/bash
DIR="/home/john/my directory/"
for f in "$DIR"/*.txt
do
  echo "${f}"
done

In the above, file names with spaces or other difficult characters will be handled correctly.

A more advanced approach could use bash arrays:

#!/bin/bash
FILES=("/home/john/my directory/"*.txt)
for f in "${FILES[@]}"
do
  echo "${f}"
done

In this case, FILES is an array of file names. The parens surrounding the definition make it an array. Note that the * is outside of quotes. The construct "${FILES[@]}" is a special case: it will expand to a list of strings where each string is one of the file names. File names with spaces or other difficult characters will be handled correctly.

While using arrays as shown by John1024 makes a lot more sense, here, you can also use the split+glob operator (leaving a scalar variable unquoted).

Since you only want the glob part of that operator, you need to disable the split part:

#! /bin/sh -
# that also works in any sh, so you don't even need to have or use bash

file_pattern="/home/john/my directory/*.txt"
# all uppercase variables should be reserved for environment variables

IFS='' # disable splitting

for f in $file_pattern # here we're not quoting the variable so
                       # we're invoking the split+glob operator.
do
  printf '%s\n' "$f" # avoid the non-reliable, non-portable "echo"
done