Spence's Job Market Signaling Game
I don't have a copy of Gibbons handy, so I cannot speak to the specific model presented there, but only generally. The intuition of the conclusion is based on the combination of the following factors:


Whenever the firm can tell the high and low types apart, it's willing to pay a high wage to high type and a low wage to low type 
If the firm cannot tell the two types apart, it will pay both types the same wage equal to their average productivity
Low type will try to pretend to be high type whenever possible to get a higher wage (note that even the average wage is higher than the low wage)
High type, in order to get the high wage as opposed to the average wage, will try to distinguish themselves from the low type whenever possible


Hence, if $e^*(H)$ is a sufficiently small cost for the low type to pay to pretend to be high type, then the high type choosing $e^*(H)$ will only result in them getting the average wage instead of the high wage. 

As a consequence, high type would be willing to pay a bigger cost $e_s&gt;e^*(H)$ --- a cost so big for the low type that they won't find it profitable to pay to get the average wage, but at the same time small enough for the high type so that their gains from getting the high wage instead of the average wage outweigh such a cost --- to distinguish themselves from the low type. 

The key here is to take into account the presence of the low type (and asymmetric info). They are the ones who cause the reduction of the high type's wage to the average level at cost $e^*(H)$.