To import a file containing multiple-value custom field data, you need to format the data in your file so the custom field values have a "double pipe" separator between them.
File type examples
Here's how a single row of subscriber data would be formatted in a comma separated values (.csv) file:
Name, Email, Interests, Country
Simon Says, simon@says.com, Surfing||Movies||Reading, Australia
Use the same formatting if your subscriber data is in a spreadsheet (see accepted file formats), as shown here:

We use the same "double pipe" formatting when you export a list with data from multi-option custom fields.
Overwritten values
By default, importing a file to your subscriber list will overwrite the values assigned to an existing subscriber's custom fields.
This is fine for many custom fields, as they only accept one value. However "multiple option (can select many)" fields can contain many values, potentially leading to a situation where multiple values are overwritten by new values imported from your file.
To keep both your old and new data for these fields, use this workaround.