One of the limitations of using a multi-select picklist field in Salesforce is that the values selected are not searchable. An idea was posted more than a decade ago on the Salesforce Idea board, but it doesn’t seem like that will happen any time soon. Here is the link to the idea if you want to vote for it: https://success.salesforce.com/ideaview?id=08730000000Bq17AAC.
The solution we recommend is creating a text area field, keep it off the layout since you may not want the users to manually edit it, and create a Trigger that adds the selected values of the multi-select picklist into this text area. The values can be comma or semi-colon separated in the text area. Even if you have multiple picklist fields, you can write all the values automatically into this one text area field.