Multiple select list box is not possible in browser based forms. What you can have is repeating section with dropdown control. And dropdown bind to dataconnection from which data has to be fetched. If you have static values you can use check box else you can use dropdown values.
Once this is done, open form options -> property promotion and select the dropdown or checkbox inside repeating table. From the dropdown in function select merge. All the selected values in dropdown or checked checkboxes values will get populated as the form library column.
Let me know if you have some question or concerns.