The problem is the f[x]= in the function to be plotted. Try it with a plain expression or define the function outside Plot. (Plot analyzes the expression passed for discontinuities, but it apparently does not handle the case when the expression is of the form Set[f[x], expr].)

Either

Plot[(1 + x^2)/(-4 + x^2), {x, -6, 6}, 
  PlotStyle -> 
   Directive[RGBColor[1, 0, 0], AbsoluteThickness[2.25], 
    Arrowheads[{-.05, .05}]], 
  GridLines -> {Range[-6, 6, 1], Range[-6, 6, 1]}, 
  Ticks -> {Range[-6, 6, 1], Range[-6, 6, 1]}] /. Line -> Arrow

or

ClearAll[f];
f[x_] = (1 + x^2)/(-4 + x^2);
Plot[f[x], {x, -6, 6}, 
  PlotStyle -> 
   Directive[RGBColor[1, 0, 0], AbsoluteThickness[2.25], 
    Arrowheads[{-.05, .05}]], 
  GridLines -> {Range[-6, 6, 1], Range[-6, 6, 1]}, 
  Ticks -> {Range[-6, 6, 1], Range[-6, 6, 1]}] /. Line -> Arrow