Put the cursor in the next cell and type Shift Return, or Enter, to run the animation.

Animate[
{pt} = Evaluate[{x[t], y[t]}/.fox]; (*pt is point at time t*)
Show[path,Epilog -> {PointSize[.015],Point[{t,1}], Point[pt], Thickness[.002],Line[{pt, {t,1}}]}, PlotRange ->{{-.03,2.53}, {-.03,1.03}}, AxesLabel -> {x,y} ],
{t, 0, 2.5}, 
Initialization->{
fox=NDSolve[{
x'[t]==((t-x[t])/(Sqrt[(((t-x[t]))^2)+ (((1-y[t]))^2)])),
y'[t]== ((1-y[t])/(Sqrt[(((t-x[t]))^2)+ (((1-y[t]))^2)])),
x[0]==y[0]==0},{x,y},{t,2.5}];
path = ParametricPlot[{{t, 1}(*rabbit*),
Evaluate[{x[t],y[t]}/.fox]},{t,0,2.5}]}]