From this, I can see two solutions. Either you could recreate the effect in Unity, thus negating the problem entirely and leading to potentially cooler results, or you could just fix and export the thing.
Reanimate in Unity
Rather than trying to get the object and its animations to export from Blender into a usable format, you could export the model and reanimate it in unity. This would not only involve slightly less hassle, but it might even create a cooler effect, since you could use real-time physics and collisions in-game.
First, export the model, and import it into Unity. You may want to parent the little pieces of the cube to an empty before or after the export so you can keep track of them more easily.
Once the model is in Unity, you could use Unity's built in physics to create the effect in real time. Add mesh colliders and rigidbodies to each piece of the cube, and make a script that launches each piece away from the cube when it needs to explode.
Just Export It!
If you really need to have the animation that you made in Blender, then you need to clean up the model some before you can export it. The current problem is that each piece of the cube isn't connected, but is instead its own separate object. This means that Blender has made a separate animator for every piece of the cube.
To fix this, you need to create an empty axis object and parent all of the pieces to this object. Then, bake the animation to keyframes on the axis object. Make sure that you remove any keyframes on the pieces before baking. This should fix the issue, so try exporting the file again.
If it still doesn't work, then you will have to get destructive with your work. You will need to delete the animations off the pieces and recreate them as a single animation, with the empty axis as the animator. Since this completely deletes the animation, you may just want to recreate the effect in Unity if you reach this point.
I would recommend recreating it in Unity, but it entirely depends on your needs. If this is part of a character's animation, or if you are on a very tight performance budget, then freely moving physics objects may not be the best option. But if you just want to use this as an interesting effect, or even if this is in a cutscene, then Unity's physics may give better results.