implicitEulerLogistic[n_]:=(
h=1/n;
y=Table[0,{n+1}];
y[[1]]=0.01;
For[i=1,i<n+1,i++,
initialGuess=y[[i]]+10h y[[i]](1-y[[i]]);
y[[i+1]]=yNew/.FindRoot[yNew==y[[i]]+10h yNew(1-yNew),{yNew,initialGuess}]
];
y
)

n=50;
y=implicitEulerLogistic[n];
plotAppr=ListPlot[Table[{i h, y[[i+1]]},{i,0,n}]];
plotExact=Plot[(0.01/(0.01+(1-0.01)E^(-10t))),{t,0,1},PlotStyle->Red];
Show[plotAppr,plotExact]

n=250;
y=implicitEulerLogistic[n];
plotAppr=ListPlot[Table[{i h, y[[i+1]]},{i,0,n}]];
plotExact=Plot[(0.01/(0.01+(1-0.01)E^(-10t))),{t,0,1},PlotStyle->Red];
Show[plotAppr,plotExact]