This walkthrough info covers it pretty well:
The aim of this puzzle is to move the white pieces to occupy the squares where the black pieces are, and vice versa. The bishops can only move diagonally and cannot move onto squares where they can be taken by the opposing color. This also means that the bishops on different colored squares cannot affect each other. Thus the puzzle is to swap the bishops standing on black squares and just solve the whites the same way.

---SPOILER---

Solution:
First, work only with the pieces on one color of squares, then use the mirror image of the moves to get the bishops on the other color across. E.g.:
White E2→D3
Black E3→D2
White A2→B3
Black A3→B2
White D3→B1
Black D2→B4
White B3→C4
Black B2→C1
White A4→D1
Black A1→D4
White D1→E2
Black D4→E3
White B1→C2
Black B4→C3
White C2→A4
Black C3→A1
White E4→C2
Black E1→C3
White C4→A2
Black C1→A3
White E2→C4
Black E3→C1
White C2→D1
Black C3→D4
White A2→B1
Black A3→B4
White B1→E4
Black B4→E1
White C4→D3
Black C1→D2
White D1→B3
Black D4→B2
White B3→A2
Black B2→A3
White D3→E2
Black D2→E3

(A,B,C,D,E are the columns from left to right; 1,2,3,4 are the rows from bottom to top)