What do exclamation marks on tiles mean?

These are tiles where explosions/area-of-affect attacks are going to take place during the next attack phase.

Examples of explosions/aoe being caused: The pixie explosion attack, mortar attacks (including rainbow horse), the mortar gun used during Helmitor battle, and the mushroom (fart attack thing).

Basically any attack where it will affect multiple tiles surrounding it and puts your own characters at risk of damage. The ! warning shows up any time that an area of affect attack will be taking place. These can damage any character, friend or foe which is why they signal it being dangerous for being in the tile.

In your screenshot, basically what it's saying is the mushroom is moving to that tile, and will damage people in the surrounding tiles.

Source: Playing the game a whole lot.

I found a Steam Forum that states that these are tiles you want to avoid. The help menu in the game describes these as "Danger".

The exclamation mark is 'look out' or 'avoid'.

A screenshot from that forum is below. "Danger" is the second one down on the second column: