MapInfo is fundamentally different than ArcGIS in the way it handles things like this. There isn't a button to only filter out selected features. Instead, what you need to do is create a temporary table on your selection.
In the below image, I have a table with 3 lines with one selected:
And you can see the selection in the browser:
Now, in the Window Menu, select New Browser Window... Or press F2.
This will open a new browser of just the selection. Notice that the name of this new table is called "Query 1".
You can manipulate data in Query 1 and it will be reflected in your main table because they're linked. Note, this query is stored in memory. If you want to maintain this query in another session, you'll need to save your workspace. You can close Query 1 like any other table.
Also note, that if you deselect the features, Query 1 is still there and you can still manipulate the data as you like and the changes will reflect in your original table.