Tiled's recent's version has changed the way they add new tilesets.

• Embed in Map - this means that the tileset is unique for that map and cannot be used elsewhere. You can't seem to animate tiles that used this function.
• Save TSX as JSON - this is similar to RPG Maker's Database -> Tileset Editor. This is useful if you need to use the same animated tiles on multiple maps.

Embed a Tileset to a Map

1. Press the New Tileset icon in Tilesets tab.

3. The New Tileset window should appear.  

4. Click Browse and select tileset from Project Folder/img/tilesets.

5. Change the Tile Width and Tile Height to your desire. Normally, it should be based on what you have set on that map. For this example, we'll be setting 48x48. It should look like this:

6. Press Save As... and save it on the folder where all your TSX files will be saved.
7. You will get a new tab of the TSX file. Do not close this tab without drawing a tile on the map first. Doing so, will make the tileset disappear from your map.

8. If you accidentally closed it, it will just require an extra step. Press Map -> Add External Tileset... and select the newly created TSX file. And you're done!

To have animated tiles, read the Setting Animated Tiles.