Unfortunately, Airtable is significantly more limited than FileMaker — Airtable can probably only do about 5-10% of what FileMaker can do. It’s like comparing a running shoe to an airplane.
Airtable gives you no way to change the records that you’re currently viewing except by using the “Filter” feature. So, to accomplish what you’re looking to do in Airtable, you would still need to work within the constraints of the filter that you have set up.
So, one way that you could approach this would be to add a new checkbox field to your table. Then, add an additional filter criteria that only shows records if their checkbox field is unchecked. Whenever you check the checkbox, the field will disappear from the view.
Of course, if you wanted that field to show up again in the view, you would need to hunt down that record on a different view and uncheck the checkbox again. So it would be a pain to get that record to reappear again. (Although you could automate something with a JavaScript or an external automation tool to uncheck previously-checked checkboxes on a schedule — or whenever some other action happens.)
This also brings up another limitation of Airtable’s filtering feature: you can’t mix “and” & “or” filtering criteria together. All of your filtering criteria need to be “and” OR all of your filtering criteria need to be “or”.