That is not a tileset in the format expected by the Terrain Brush. This tileset is expected to be used in RPG Maker, which implements a very specific way of cutting the tiles into pieces and then recombining them in various ways.
There is an issue open about adding support for RPG Maker autotiles to Tiled, where you can also find some information and links that may help set up this tileset for use with the Terrain Brush:
To be able to use this with the Terrain tool, you’d need to use this as an 8x8 tileset instead of 16x16, so that you can access each tile corner as a separate tile.
(The sides of the transition tiles aren’t labelled as dirt since those tiles are largely identical to your main dirt tile pieces.)
A quickly terrain-brushed map:
As you can see, as 8x8, the dirt doesn’t quite tile, so you’d need to tweak it if you want to work this way.
If you must use this as 16x16 (e.g. if your other tilesets are also 16x16 and are a hassle to use as 8x8, you’d need to modify this tileset to have all the corners Tiled’s Terrain tool requires, which is very similar to what RPG Maker does under the hood (it builds all the corners out of the sub-tiles in the tileset).