Use separation of variables to find a solution $u= u(x,t)$... 
So I get up to the last paragraph of the solution. I can get the bases of the solution, but beyond that, I'm really confused as to what they did. Any help would be appreciated!

Usually, the next step demands to develop the initial condition for $t=0$ in terms of the solutions for $X(x)$ as a linear combination or as a weighted integral over the parameter $c$. 
However, in this case we are "lucky" that the initial condition itself is already such a solution. Thus $\pm\sqrt{c-1}=1$ resolves to $c=2$. Up to this point, the full solution is a linear combination of $e^{\pm\sqrt2\,t}·e^x$, the condition to fall to zero excludes the growing function $e^{\sqrt2\,t}·e^x$, leaving $e^{-\sqrt2\,t}·e^x$ as solution.
