The logistic equation is given by $$\frac{dP}{dt} = rP(1-P)$$
With the forward Euler method it can be intuitively discretized to $$P(t+h)-P(t) = hrP(t)(1-P(t))$$
Setting h=1 and $P(t) = P_{n}$ one obtains the following form. $$\Delta P_{n} = rP_{n}(1-P_{n}) $$
Which is what an intuition behind the logistic equation says: the change in a population P is proportional to P with rate r (resulting in exponential growth), but is hindered by the carrying capacity K=1 giving an additional term proportional to $-P^2$.
The logistic map is given by $$P_{n+1} = \alpha P_{n}(1-P_{n}) $$
Setting $\alpha = 1+r$ is a way to end up at the differential equation again, but the interpretation becomes a different one.
If we try to squeeze the $P_n^2$ into the parenthesis, we can obtain
$$P_{n+1}-P_n=rP_n(1-P_n-{P_n\over r})$$
Now, we try to squeeze LHS's $P_n$ into the parenthesis, w obtain:
$$P_{n+1}=rP_n(1+{1\over r}-P_n-{P_n\over r})$$
Now, if we try to move the $r$ terms out, we can find this neat equation:
$$P_{n+1}=r\left(1+{1\over r}\right)P_n(1-P_n)=(r+1)P_n(1-P_n)$$
Since $r$ is some constant, we can now let $k=r+1$, so the logistic equation becomes logistic relation:
$$P_{n+1}=kP_n(1-P_n)$$
I hope this will help you.