Actually, RESULT contains what you want — to demonstrate:

echo "$RESULT"

What you show is what you get from:

echo $RESULT

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).

Parsing multiple output

Introduction

So your myscript output 3 lines, could look like:

myscript() { echo $'abc\ndef\nghi'; }

or

myscript() { local i; for i in abc def ghi ;do echo $i; done ;}

Ok this is a function, not a script (no need of path ./), but output is same

myscript
abc
def
ghi

Considering result code

To check for result code, test function will become:

myscript() { local i;for i in abc def ghi ;do echo $i;done;return $((RANDOM%128));}

1. Storing multiple output in one single variable, showing newlines

Your operation is correct:

RESULT=$(myscript)

About result code, you could add:

RCODE=$?

even in same line:

RESULT=$(myscript) RCODE=$?

Then

echo $RESULT $RCODE
abc def ghi 66

echo "$RESULT"
abc
def
ghi

echo ${RESULT@Q}
$'abc\ndef\nghi'

printf "%q\n" "$RESULT"
$'abc\ndef\nghi'

but for showing variable definition, use declare -p:

declare -p RESULT RCODE
declare -- RESULT="abc
def
ghi"
declare -- RCODE="66"

2. Parsing multiple output in array, using mapfile

Storing answer into myvar variable:

mapfile -t myvar < <(myscript)
echo ${myvar[2]}
ghi

Showing $myvar:

declare -p myvar
declare -a myvar=([0]="abc" [1]="def" [2]="ghi")

Considering result code

In case you have to check for result code, you could:

RESULT=$(myscript) RCODE=$?
mapfile -t myvar <<<"$RESULT"

declare -p myvar RCODE
declare -a myvar=([0]="abc" [1]="def" [2]="ghi")
declare -- RCODE="40"

3. Parsing multiple output by consecutives read in command group

{ read firstline; read secondline; read thirdline;} < <(myscript)
echo $secondline
def

Showing variables:

declare -p firstline secondline thirdline
declare -- firstline="abc"
declare -- secondline="def"
declare -- thirdline="ghi"

I often use:

{ read foo;read foo total use free foo ;} < <(df -k /)

Then

declare -p use free total
declare -- use="843476"
declare -- free="582128"
declare -- total="1515376"

Considering result code

Same prepended step:

RESULT=$(myscript) RCODE=$?
{ read firstline; read secondline; read thirdline;} <<<"$RESULT"

declare -p firstline secondline thirdline RCODE
declare -- firstline="abc"
declare -- secondline="def"
declare -- thirdline="ghi"
declare -- RCODE="50"