It doesn't work for the same reason that

ClearAll[a, b, x, f]
f[x_] := (a x^2)/(b + x)
With[{a = 10, b = 5},
 Plot[f[x], {x, -10, 10}, PlotRange -> {-20, 20}]
 ]

doesn't work. Both With and Manipulate attempt to make a replacement of the variables a and b with numeric values before running the plot command. But before evaluating f[x], there is no a or b to replace! That's why you had to have a replacement rule inside the plot command before in order to get it to work. In your plot command, it sees f[x], then makes the replacement rule, then plots it. Further, there is probably an issue as to whether Manipulate scopes like Block or like Module.

You could do this in a number of ways. Most obvious, you could modify the definition of f to take three arguments, as in f[x_,a_,b_] := (a x^2)/(b + x). But you could also do what you were doing before, with a replacement rule inside the plot command, but it can't refer to a as the manipulation variable, you need a dummy variable, which I just call aa

Manipulate[Plot[f[x] /. {a -> aa, b -> bb},
  {x, -10, 10}, PlotRange -> {-20, 20}],
 {{aa, -4, "a"}, -4, 4}, {{bb, -4, "b"}, -4, 4}]