What is this 3.5mm plug designed for?
The ends of wires in a cable get soldered to the left end of that assembly. Then the assembly is placed into a mold and hot plastic is injected around the wire to form a handle for the end of the cable. The plastic molding often includes a flexible strain relief for the cable. 

Typical result:



Often times the best way to understand something is to take it apart. Here I show the wire attachment to a miniature jack by having cut the cable molding away. You can see that the wires are simply solder tacked onto their appropriate pads. Note that the back ends of these things will vary from manufacturer to manufacturer.