The convolution term does not form part of the state transition matrix. The state transition matrix comes about purely by the initial response x(0), and not the input u(t).
Next, as I have already mentioned, tau is how much one function has been shifted in relation to the other during convolution. In this case, the two functions are u(t) and exp(at) respectively. In convolution, the limits of the integral are +-infinity, so tau will assume all these values during integration.
I guess many texts have taught you how to do convolution, but the principles behind it is really quite intuitive. You jab a billard ball, it responds by rolling, EVEN AFTER the jabbing has stopped. If you see the jabbing as an impulse input, so now imagine that instead of jabbing, you give a continuous input. If the system is linear, the law of superposition applies and the continuous input can be seen as a summation of impulse inputs. If each impulse input gives you the impulse response, you just have to sum the impulse response of all the impulse inputs, ie integrate and shift, that is your convolution.
In the derivation of the state transition matrix, the impulse response is exp(at). The continuous input is u(t). So the response due to u(t) is the convolution of u(t) and exp(at).