Regardless of how the values are entered on the form (e.g. multiple rows in a table, with the names ending in [] so you have an array of values, vs putting in a textarea and somehow parsing that), you will need to loop through the values and create multiple INSERT queries to add the rows to the child table.
Ideally, you would wrap the whole thing in a transaction, but if you are using some versions of mysql that isn't possible.
Also, I assume you are assigning some kind of unique ID automatically to the parent row: you can retrieve this value (using mysql_insert_id() or SELECT
currval(sequence) for postresql) and use it to identify the child-table rows.