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

Manipulate[
y=explicitEulerLogistic[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];
GraphicsRow[{Show[plotAppr,plotExact,PlotRange->All],
ListPlot[Table[{i h,Abs[y[[i+1]]-(0.01/(0.01+(1-0.01)E^(-10i h)))]},{i,0,n}],PlotRange->All]}]
,{n,4,800,10}]