When you first study Special Relativity you meet the strange phenomena of time dilation, length contraction and the 'twins paradox'. While it's true that the equations of the Lorentz transformation give the right answer, they may fail to provide insight into what's really going on.
The true meaning of special relativity is visual and geometric. R. S. Zimmer wrote a wonderful book called "Getting the picture in special relativity" but it is sadly out of print. However, he also wrote a shortened version for the Open University course "Space, Time and Cosmology" (Block 2) and this covers the main ideas in a beautifully clear way.
Download his ten page article here.