$E_x$ is the expected time to roll the first $5$. When you throw a dice, it's either going to be $5$ with the probability $\frac{1}{6}$ and the rest with the probability $\frac{5}{6}$.
Once you have thrown the dice, for the first time, you add $1$ to $E_x$ (The first term, $+1$). At the first roll, as I said, it's going to be $5$ with the probability $\frac{1}{6}$. But then you don't have to throw anymore, so there is nothing to add to $E_x$(That's $\frac{1}{6}\cdot0$ ). Lastly, If you didn't throw $5$ at the first roll with probability $\frac{5}{6}$, you then start the process from the beginning. In other words, since we have added $1$ in the equation at the beginning, which counts for the first throw, we can just think the second throw as a first throw of a new process to count $E_x$. Therefore, with the probability $\frac{5}{6}$, we start this process again. (The last term $\frac{5}{6}E_x$)