One could say that the Lord of the Rings films have a heroic struggle that adults can relate to. However, you can see equally titanic struggles and quests on many Saturday morning cartoons. So a "hero journey" or a titanic struggle isn't enough to make a story appeal to an adult.
In Lord of the Rings, we also see adult or complicated emotions. For instance, Aragorn (also known as Strider) feels compelled to aid in the destruction of the ring out of guilt. His ancestor was responsible for the failure to destroy the ring the first time around.
And in The Two Towers, the elves have the difficult decision whether to help rescue mankind for a fate that mankind brought upon itself by failing to destroy Sauron's ring when it had the chance.
Also in The Two Towers, Gollum is torn between thinking and acting like the degraded creature he has become, or the human (Smiegel) he used to be.
In both films, there's an elf woman, Arwen, in love with Aragorn. She maintains her love, even though she knows she's immortal and he's mortal, and it will bring her sorrow in the end when time takes him away from her.
All of these issues are ones that have the kind of complexity that can appeal to an adult.