Try Artin's Geometric Algebra.
You might also find interesting Shafarevich's book “Basic Notions of Algebra”, for understanding the philosophy behind algebra. I strongly recommend that you spend some time with that book.
Also try Coxeter's various books such as Projective Geometry, etc..
But of course if you really want to understand the "geometry behind the algebra", then you should first have a good footing in algebra, and then you should study algebraic geometry. Even for Artin's Geometric Algebra, you better first have an understanding of algebra before reading it. It appears to me that at the moment the best bet for you will be Kostrikin-Shafarevich.