Why line graph is a proper subclass of claw-free graph?
Your question is answered on Wikipedia:


  "Line graphs may be characterized in terms of nine forbidden subgraphs;[2] the claw is the simplest of these nine graphs."


The reference is to the following paper:


  [2] Beineke, L. W. (1968), "Derived graphs of digraphs", in Sachs, H.; Voss, H.-J.; Walter, H.-J., Beiträge zur Graphentheorie, Leipzig: Teubner, pp. 17–33.


Wikipedia contains a pictorial description of these nine graphs. One of them is $K_5^-$, obtained from $K_5$ by removing a single edge.

If you take any of the other forbidden subgraphs then you get a claw-free graph which isn't a line graph.