(**Migrated** by request from the comments.)

Bruhn and Schaud's (2013) **The journey of the union-closed sets conjecture** provides a rather readable write-up. Particularly relevant is the section **Obstacles to a proof**; for example, you may check just after *Conjecture 15* in which the authors ask (essentially) your question here:

"So, why then has the conjecture withstood more than twenty years of proof attempts?" (p. 14)

Bruhn and Schaud then list three possible techniques of proof, and go into a bit of detail around why they do not seem to work out; these techniques are: *injections*, *local configurations*, and *averaging*.

The paper also provides a few relevant re-formulations using, e.g., lattices, (maximal stable sets of bipartite) graphs, and the "Salzborn" formulation (p. 12). In each case, a re-formulation of the **Frankl** (or **union-closed sets**) **conjecture** brings corresponding ideas and techniques with varying potential; the authors of this particular survey do well by their promise early on:

"The focus of this survey is on the methods employed to attack the conjecture. Our treatment of the literature is therefore somewhat uneven. Whenever we can identify a technique that, to our eyes, seems interesting and potentially powerful we discuss it in greater detail" (p. 3).