Have you covered eigen-values and eigen-vectors? (This is required knowledge to solve a linear ODE system).
Basically the (a1,b1)^T and (a2,b2)^T are the eigen-vectors and the lambda's are the eigen-values for the solution to the DE.
If you want more information, you'll have to either pick up a linear algebra book or a book on DE's and look at the proof.
In Linear Algebra there are techniques to solve differential equation relationships with operators and when you deal with operators and matrices then you need to use the results of linear algebra which involve eigen-decomposition.
I don't know the proof myself, but the key ideas will involve functions of an operator (operator algebras) and the differential and integral calculus on operators.
The general proof is not going to be easy.