A low tech-skilled volunteer here.
Once you’ve matched files to track and clicked Save, the files should now be tagged and in your storage.
To check that the tags have been applied, select a file/track and then look in the bottom Tag Display Pane - it should now have “New Values” being the same as “Original Values” column. The image below shows an instance where New and Original values are different.
“Once you’ve saved an album and/or want to remove it from view, right click the album in the right-hand pane and click ‘Remove’.” (https://picard.musicbrainz.org/docs/basics/)
Does that answer your question?
(Picard also has options to write the saved tagged files to new folders, with new names, if that is what you want. One method to tag a collection is to make a back-up of the collection, then use Picard to tag and move folders out of that back-up to a New Tagged Music folder, and also have Picard, at the same time, delete the albums which it tags from the back-up folder. This separates your tagged albums from your untagged albums.)
Do keep asking, tagging is more complex than it sounds. If you get stuck for more than 5 minutes - ask.