This is a tough one, because the lyrics do not always exactly correspond to complete thoughts or complete sentences. It may be helpful to look at other lyrics in "Something", to understand exactly what the author is saying.
Something in the way she moves
Attracts me like no other lover
- The author does not know exactly what causes him to be attracted to her, more than any of his other "lovers". He knows that it is related to how "she moves". It could be the way she walks, holds her hands, or dances.
Something in the way she woos me
- There is something special, or awesome, about the way that "she woos me" (she tries to gain his love), but the author does not know exactly what.
Somewhere in her smile she knows
That I don't need no other lover
- The author knows that she knows that he only needs her, and he recognizes this concept through the way that she smiles. Perhaps her smile is confident or seductive.
Now let's look at your verse.
Something in the way she knows
And all I have to do is think of her
Something in the things she shows me
I don't want to leave her now
You know I believe and how
Something in the way she knows : I believe this is a reference to "Somewhere in her smile she knows", how she knows that he is interested in her.
Something in the things she shows me : This is more vague. "The things she shows me" could be her smile, the way she talks, the way she moves, or any other quality that the author mentions in the songs.
The most important takeaway: the author does not seem to form a complete thought. Rather, he seems to be listing her qualities as he thinks about her.