Capturing multiple line output into a Bash variable

In case that you're interested in specific lines, use a result-array:

declare RESULT=($(./myscript))  # (..) = array
echo "First line: ${RESULT[0]}"
echo "Second line: ${RESULT[1]}"
echo "N-th line: ${RESULT[N]}"

Another pitfall with this is that command substitution — $() — strips trailing newlines. Probably not always important, but if you really want to preserve exactly what was output, you'll have to use another line and some quoting:

RESULTX="$(./myscript; echo x)"
RESULT="${RESULTX%x}"

This is especially important if you want to handle all possible filenames (to avoid undefined behavior like operating on the wrong file).

Actually, RESULT contains what you want — to demonstrate:

echo "$RESULT"

What you show is what you get from:

echo $RESULT

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As noted in the comments, the difference is that (1) the double-quoted version of the variable (echo "$RESULT") preserves internal spacing of the value exactly as it is represented in the variable — newlines, tabs, multiple blanks and all — whereas (2) the unquoted version (echo $RESULT) replaces each sequence of one or more blanks, tabs and newlines with a single space. Thus (1) preserves the shape of the input variable, whereas (2) creates a potentially very long single line of output with 'words' separated by single spaces (where a 'word' is a sequence of non-whitespace characters; there needn't be any alphanumerics in any of the words).