Your code looks good, and you've definitely normalized the histogram correctly. However, you've got one thing that needs fixing. The random values, p, in your probability density function should be the same random values. You have generated them independently, but p only need to be randomly generated once. I've made a simple fix to your code and commented on the lines!
maxiter = 10^5;
x_limit = 15;
x_res = 0.01;
% Generate vector of uniform random numbers
randVec = rand(maxiter,1);
% f(p) = ln(p/(1-p)) => PDF random values utilize same p vector
X = log(randVec./(1-randVec));
x = -x_limit:x_res:x_limit;
[N, BIN] = histc(X, x);
Nnorm = N./trapz(x,N);
subplot(2,1,1)
h = bar(x,Nnorm, 'hist'); hold on;
plot(x, exp(-x)./(1+exp(-x)).^2,'r--')
title('PDF for logistic distribution')
hold off
A1 = trapz(x, exp(-x)./(1+exp(-x)).^2);
A2 = trapz(x,Nnorm);
subplot(2,1,2)
bar(x, cumsum(Nnorm*x_res)), hold on
plot(x,exp(x)./(1+exp(x)),'r--')
title('Cumulative PDF for logistic distribution')
