Why shud one use convolution ?Inorder to get the state transition matrix

from

x'=Ax+Bu

x(t)=exp(at)x(0) + integral[exp(a(t-tau)) bu(tau) dtau

where tau is the variable of convolution.

i have read that convolution is used when integration is reqd,for instance,in capacitors and inductors which dont respond to an input immediately.....but can someone PLEASEEEEEEEEEE help me understand how convolution really works?

Thanks for any help always!