The first equation – in which you omitted $\psi$ on the left hand side – is a differential equation for $\psi$. The second equation reflects the fact that the equation is linear, so the evolution of $\psi$ after time $t$ will have to be linear as well, and it is given by an operator called $S(t)$ that stands on the left side from $\psi$, the most general linear operator acting on state vectors.
The particular form of $S(t)$ in the third equation is a solution to the differential equation, i.e. the first equation you wrote. Solutions to differential equations can't always be "derived in a straightforward way" and even if they can be solved, there may always be some kind of a guess. Anyway, everyone familiar with basics of calculus sees why this $S(t)$ solves the differential equation at the top. The $t$-derivative of the exponential is once again the exponential times the $t$-derivative of the argument of the exponential, and the derivative of the argument with respect to $t$ is clearly $-i H_f/ \hbar$ which is the right factor that jumps out in the first equation.
Concerning the $\tau_3$ equation, you would have to explain what you exactly mean by $\tau_3$ and what's the argument of $S^\dagger$. In general, however, the equation says that $S$ and $S^\dagger$ are equivalent. What "matrix manipulation" do you exactly want to see?