Hi Ushanka;
The issue is that the letters are not manifold. The back surface of them that intersect with the apple are not closed. And they are rather jagged as well.
In addition, the bottom of you stem is not closed either.
I am becoming a big fan of Meshlab, but can't tell you how to fix it with that quite yet.
Walking you through fixing this in Blender would be a Tutorial all its own. So I'll just explain the basic gist.
the stem I just closed by extruding the edges to a point, and then removing the doubles.
The letters, I separated from the model making them there own mesh. Moved them to a separate layer. Then looking down (Num 7 in blender) I deleted all the verts beyond the front faces of the letters. I.e. Reduced the letters to only they're 2D faces. And then extruded them along the normals. I also hollowed out your apple as it would cost $330+ USD to print as a solid object.