I am unable to save product multiselect values .I tried both ways Here is code
Way 1
$objectManager = \Magento\Framework\App\ObjectManager::getInstance();
$_product = $objectManager->create('Magento\Catalog\Model\Product');
$_product->load(757);
$string = '5641,5642'; // option ids
$_product->setData('multi_select_color_code',$string);
$_product->save();
Way 2:
$objectManager = \Magento\Framework\App\ObjectManager::getInstance();
$_product = $objectManager->create('Magento\Catalog\Model\Product');
$_product->load(757);
$array= array('5641','5642'); // option ids
$_product->setData('multi_select_color_code',$array);
$_product->save();