World Craft! Loved it! I had a little moan way back about how snapping a vertex to an edge would create a new vertex and physically join them in World Craft, but in Blender this is a multiple step process involving subdividing the edge, sliding the vertex and merging the vertex you wish to join.
This in essence is probably your problem. As aomeoni stated, this will create a non-manifold mesh (non watertight) resulting in big phat holes in your mesh. In Unity these parts with holes wont render.
In Blender, where you have snapped a vertex to an edge, grab a vertex and move it around with G. You will see that a big hole appears in your solid. Create a vertex on the edge by subdividing and Vertex-Slide it to where your snapped-to vertex is. Enabling Auto-Merge editing will make this easier. When you have eliminated all these holes, recalculate your Normals to Outside.
Another thing that makes something non-manifold is Internal edges and faces. Where you have all those edges on the curve of your axe, internalised by the straight edge, delete the internal edges.
Another thing you must do is Unwrap your mesh to create a UV map.
Note that Unity will NOT import materials from Blender so you must texture the model externally (or export your textures separately) and import the textures in to a Unity shader, which you select in the Model’s Unity material.
I must say, it looks a bit like you imported to Blender from Sketchup. Sketchup is not very mesh friendly and exported meshes are an absolute effin mess. Not good for Unity.
Also, in your export screen, deselect Camera and Lamp, as you don’t want those being imported in to your Unity scene.
But as for the axe, really, make a new one! It’d be quicker than fixing this one.