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What you did wrong here is the part date('d', $rem_days). The function date() should be used to convert timestamp to a formatted date, not to converting a time difference in timestamp to time difference in days.

You can fix this by replacing date('d', $rem_days) with floor($rem_days/86400).

The complete code should be:

$event_date = strtotime( get_field( 'event_date', false, false ) );
$curr_date  = time();
$rem_days   = $event_date - $curr_date;
if ( $rem_days <= 0 ) {
    $event_msg = '<strong>Event Expired</strong>';
} else {
    $event_msg = '<strong>' . floor( $rem_days / 86400 ) . '</strong> Days Remaining';
}

Note: As per Pat J's comment, WordPress provides a bunch of handy {timespan}_IN_SECONDS constants in wp-includes/default-constants.php, including DAY_IN_SECONDS.

So in the above code, floor( $rem_days / 86400 ) can be replaced with floor( $rem_days / DAY_IN_SECONDS )