A Postnikov system of a path-connected space $X$ is an inverse system of spaces
$$ \cdots \to X_{n}\xrightarrow {p_{n}} X_{n-1}\xrightarrow {p_{n-1}} \cdots \xrightarrow {p_{3}} X_{2}\xrightarrow {p_{2}} X_{1}\xrightarrow {p_{1}} \{*\} $$
with a sequence of maps $ \phi _{n}\colon X\to X_{n}$ compatible with the inverse system such that
The map $ \phi _{n}\colon X\to X_{n}$ induces an isomorphism $ \pi _{i}(X)\to \pi _{i}(X_{n})$ for every $ i\leq n$.
$ \pi _{i}(X_{n})=0 $ for $ i>n$ .
Each map $ \displaystyle p_{n}\colon X_{n}\to X_{n-1}$ is a fibration, with homotopy fiber $F_{n} $, which is the Eilenberg–MacLane space, $ K(\pi _{n}(X),n)$.
The first two conditions imply that $ \displaystyle X_{1}$ is also a $ K(\pi _{1}(X),1)$-space. More generally, if $\displaystyle X$ is $(n-1)$-connected, then $ X_{n}$ is a $ K(\pi _{n}(X),n)$-space and all $ X_{i} $ for $ i<n $ are contractible.
continuing in wikipedia is moreover stated that the homotopy class of fibration $p_n:X_n \to X_{n+1} $ comes from homotopy class of the classifying map for the fiber $ K(\pi _{n}(X),n)$. Why? I not see at this point any direct connection. How the homotopy class of $p_n$ can be constructed resp. related to certain class of a classifying map
$$ X_{n-1} \to B(K(\pi _{n}(X),n)) \simeq K(\pi _{n}(X),n+1) $$
as clamed in wikipedia? In other words why the homotopy class of $p_n$ can be considered as a homotopy class in $[X_{n-1},B(K(\pi _{n}(X),n))]$? Could somebody unravel where this identification comes from?