That means if there is some blood shed, you become fiercer in the fight. So it makes your rage stronger! "Grease up" also means "to bribe", not in this case though. E.g. I have greased up my girl-friend with flowers, so she is bound to give me a "yes" answer to go out with her. Hope, it helps. ;)

Grease is a slippy chemical. You grease a door hinge, to make it work better, the 2 parts slide against each other better with grease in between. Grease muscles up, is a metaphor, for making the muscles work better.

I checked the quote, and I see it's from The Vampire Diaries. So that explains that the "grease" must be internal (though I really liked Jesse's explanation). The vampire Mikael had just been woken up from a slumber and, like an unused machine, probably felt a little stiff. So Kat suggests that Mikael feed on some blood to renew and refresh his body, using a "rusty machine" analogy: grease will make the parts work more smoothly (ie. Craig's explanation). So in this context, "grease up" means "give those working parts some 'grease'(blood) to make them work better." The "up" part of the phrasal verb hints at a desired result, eg. sufficiently lubricated.

When in a fight, being wet or greasy helps punches slide off. Dry skin has more friction so being hit hard is easier - a bad thing. Getting bloody can actually protect you from some of the future hard punches.