If the marginal cost is equal to 1, how does that imply marginal cost is equal to marginal benefit? The function below is a utility function simplified after subject to an implied participation constraint. 
$$
E\left(\pi_{n}\right)=e^{*}-E\left(s^{*}\right)=e^{*}-c\left(e^{*}\right)
$$ where $ \pi_n $ is net profits, s is salary, $ e^*$ is optimal effort and $ c(e^*) $ is the personal cost (disutility) of that effort

The function is maximized for $ e^* $ satisfying the first-order condition
$$
c^{\prime}\left(e^{*}\right)=1
$$
The text goes on to state that at the optimum, the marginal cost of effort, $c^{\prime}\left(e^{*}\right)$, equals the marginal benefit, 1. 

Q: Where did marginal benefit just come from? I could be just forgetting some basics, but I don't recall MC = 1 as an implication that MC  =  MB?
You have to look at this from the derivation of the profit equation.

From the equation $\mathbb{E}(\pi_n) = e - c(e)$, you can see that the marginal benefit of increasing $e$ is equal to 1. That is, for each extra $e$ you put in, you get that exact amount back in terms of expected profits. The $e$ term is the benefits, and $\frac{de}{de} = 1$ is then the marginal benefits. The same goes for marginal cost with the $c(e)$ term.



Follow up

In any objective function, labeling things "benefits" or "cost" could be tricky. Here, one could assume, from the functional form, that the function looks like $\pi = TR - TC$. It's then quite natural to label $TR = e$ and $TC = c(e)$. The benefit is the revenue you get, and cost is, well, cost.

It could very well be that, actually, $TR = e + f(e)$ while $TC = c(e) + f(e)$, where $f(e)$ could be any well-behaved function. The "correct" marginal benefits would then be $1 + f'(e)$, while marginal cost would be $c'(e) + f'(e)$. At the end of the day, though, you'd still have

$$1 + f'(e) = c'(e) + f'(e) \qquad\Leftrightarrow\qquad 1 = c'(e)$$