how to assign value to a multi picklist field using apex.

For anyone who needs to assign a Set or List of strings to a multiselect picklist you can use String.join(listOfString, ';')

Example:

List<String> listOfMultipicklistValues;
// assign to a multipicklist field
sObject.multiPickField__c = String.join(listOfMultipicklistValues, ';');
// FYI: String.join() method will merge the elements in the list and separate them with the separator in the second parameter