characteristic values or eigenvalue of a matrix

I can't understand the reasoning about the solution:

Can you help me explaining it in more detail?

explaining what? just the highlighted part? it is basic arithmetic:

xdot = A*x

where xdot = lambda*x

therefore we can write

lambda*x = A*x
lambda*x - A*x =0
note that lambda is some proportionality scaler (which we call eigenvalue) er can write
[lambda*I -A]x = 0

https://en.wikipedia.org/wiki/Eigenvalue

Hi, I need to explain why we have dx/dt = λ*x.

Hi,

That's because this is the state space representation of the equations.