Here is an example of how to move products (this example add categories to existing products). Note: Remember that when you load the Move products (tab) by default the products in Uncategorized category (or your default category) will be loaded, this is done to make it easy to see those products and move them to the correct category.
First select in the left products from (categories) the products you want to do changes with (add, replace or delete categories on), then if you like filter the products listed in the middle section using the search / filter input box.
When you found your selection, click the check products (uncheck the ones you do not want to do something with). Then in the right section (move products to), check the categories you want to add, replace or delete. In this example we want to add two categories.
Click the add categories button, the categories selected will be added and the page will refresh.