How to use Rules to set a value for a multi-value field?
Looks like you could just use the "Add to list" Rules Action. And since your question is about a multi-value field, for sure you should NOT use a  Rules Action like "Set a data value", which is used to set assign a single value to a field (which seems to be what you've tried, a common mistake when using the Rules module for such fields).

Refer to the answer to "How to set multiple values to an unlimited Term Refence field with Rules" for "some" more details ...

PS: I'm assuming this question is about D7.