I think it's where one person randomly gets the ability to play as a "Hale", which is basically a player class with super strong attacks and they have to kill everyone else (and the non-hale players have to try to defeat the Hale).
And I think if they get damaged enough, they get "max rage", which basically boosts their attack power even higher.
Also, while many Hales have custom skins and/or colour palettes, there's the odd choice of them having default skins AND palettes. (I died quickly in a match that had 2 Hales that looked just like Geminiman with the Mega Buster palette!)
To be more precise...
Saxton Hale is a mode which 1 or 2 players become bosses. The boss often has a short, melee range attack that either deals massive damage or kills in one hit (unless you're cave, Morshu, or curly & quote then you get a long range weapon).
As the boss(es) take damage, their rage meter fills, enabling the boss(es) to use a unique special attack unique to the boss (example, Saxton get super armor plus insane armor buff). They also get a super jump that enables them to reach hard to get players. The whole buster colored boss thing is a bug. Just know who the boss is to figure out that boss' skin.
EDIT- Ninja'd damn you.