The textures don’t have alpha transparency in them.
For leaf texture, the background is black. Either edit the black parts to have Alpha 0, make a greyscale alpha mask, or use nodes to get that mask. The material also needs to have transparency on and alpha and specular set to 0, so that the texture controls the transparent and opaque parts of the mesh.
Edit: here’s a node setup to add the alpha to the texture itself. Adding perfect alpha to an image that doesn’t have one is not so straightforward though because of antialiasing. The colors in the leaf edges change as they progress towards the background color.
In this it doesn’t matter much because single leaves probably aren’t seen close up.
Also, object scale is not uniform on either of the objects in the scene. This makes the UV unwrap result to something else than what is expected. Blender warns about that in the top header but it’s easy to miss.
Properties panel (N) to see the scale, and ctrl+a -> scale to apply scale.