Why does our face get red during anger?
The comments and answer so forth are mainly related to the visibility and possible social and evolutionary explanation of the 'red face'. 

While not discussing the social or evolutionary factors, I can give an overview of the biochemistry involved. 

Anger and associated feelings such as stress are part of the 'fight or flight' response. This response is regulated mainly by the release of epinephrine and norepinephrine (the same as adrenaline and noradrenaline) into the blood stream by the adrenal glands, and from neurons into the synapses. 
Wiki:Fight or Flight response
(Hence, the signal is send out both in the whole organism, and locally between neurons). 

These hormones/neurotransmitters have a number of physiological effects:


They trigger a combination of vasoconstriction and vasodillation  in different parts of the body, meaning that the blood vessels constricted and widened, respectively, in different tissues. 
This allows for a more effective transfer of nutrients to the muscles and brain. Wiki: Epinephrine
They trigger the increase of blood sugar by breakdown of glycogen deposits. Again, the body is in a situation where it needs to use the available energy ressources.
The heart rate is increased. Again, this allows for more effective transportation.


As listed above, the hormones related to anger both trigger the constriction and dilation of blood vessels in different tissues. Thus, the routing of the blood through the system also changes. 

However, the routing of the blood seems to differ, depending on the exact feeling, where blood sometimes are routed to the surfaces areas of the skin, and sometimes away from them (by means of vasoconsttriction and vasodillstion, respectively). 

Intuitively, this is can be a way of controlling the body temperature, as the body can expose the warm blood to the surroundings to a lesser or greater extent. 

I don't think it is entirely understood, what determines this, but a guess could be that in situations of anger, the body has to be able to effectively use the muscles and fight, and thus also needs to be cooled, in order for the systems to work properly. 
In situations of fear you might to hide instead for a long time and preserve the body heat and energy (just speculations). 

Besides, it also could have a social impact. 

But when blood is transferred to the facial areas, including the brain which needs energy to think quickly, and the skin surface for cooling down, it can have an affect on the perceived color of the face. 

As noted by others, the degree of visibility is highly personal, as it depends on several factors such as the degree of hormone release, hormone sensitivity, anatomical features of the face such as placement of blood vessels, skin color and so forth.