That is all done with JavaScript. In ACF Pro if you look in the file /plugins/advanced-custom-fields-pro/assets/js/acf-input.js there is a function starting on about line 7164.
Basically, from what I understand it looks for a tab field and then looks for the next tab field, checking for end points along the way, and hides or shows fields based on that. It adds several classes to the fields so it knows where the first field in the tab is and where, etc. The hiding and showing is all done by adding or removing these classed from the field wrappers.
