Merriam Webster defines rage as a violent and uncontrollable anger, a fit of violent wrath, insanity, and passion. Protests have become chaotic and by naming these outbursts as "days of rage," journalists almost give protesters the right to run wild. The uncontrollable aspect of rage seems to imply danger to participants involved in protests. While passion would be a safer word choice, rage gives the cause urgency and magnitude. Passion could be applied to positive and negative events. Rage takes a strong position towards unrest and unhappiness. Not all protests have to be violent, but to get any point across rage is a more effective tool.
Rage is also a good word to associate with protests because of the way it sounds/flows. Phonetic intensives, or sounds that connect and add to a words meaning, can be applied to the word rage. The R sound is refered to as "liquid" and "euphonious" while the harsher G at the end of the word implies cacophony. Cacophony is definitely characteristic of protests, but the R is also important to make the word catchy/roll of the tongue. This might be a stretch, but even the shift from smooth to fragmented exemplifies the process of protesting: you start calm and get more motivated.
Now to tackle the day part. Day might seem only explanatory, especially since weeks of rage have even been proclaimed, but the brevity of the period of time is important to note. A day implies a short lived protest where people can be lawless and free to oppose the government. I emphasize the word short. A day of protest is a time when angry citizens can vent, but it is imperative that violence is promptly ended so that more time can be focused on reshaping governments etc instead of calming down enraged crowds.
In my opinion, violence won't create any solutions for corrupt governments, but at least protesters are getting their opinions heard and changes are starting to be made. In countries where the average person's voice can't be heard by local government officials, days of rage may be the only option to call attention to shared concerns.