Why does Alan Brinkley describe “Is this a Republican form of government?” as “biting”? This is a political cartoon in the textbook by...

I assume that you are talking about the Thomas Nast
cartoon that I have linked to below.  It is a cartoon about the conduct of the South
during Reconstruction.

The reason that Brinkley calls this
"biting" is because it is clearly an angry cartoon that grabs a hold of a person and
makes them think.  This is the definition of "biting."

In
the cartoon, Nast is saying that the behavior of vigilante groups in the South (like the
KKK or the "White Liners" that you see referred to in the background) is unAmerican. 
The South tended to argue that Reconstruction was bad because it denied them a
republican form of government (guaranteed in the Constitution).  Nast is rebuking them,
saying that what they do to the blacks is in no way consistent with republican values. 
He is telling them that if they don't act in a republican way towards blacks, they don't
deserve to be treated as republicans themselves.

So this is
a biting cartoon because it is emotionally powerful and it has a very clear and
compelling point.