= Solution
A <Serre fibration> has the <homotopy lifting property> for disks: for every $k\geq0$, a map $D^k\to E$ and a <homotopy> $D^k\times I\to B$ starting at its composite with $p:E\to B$ admit a compatible lift $D^k\times I\to E$. Equivalently, it has that lifting property for <CW complexes>. Relative lifting for CW pairs follows by attaching cells.
Let $i:F\hookrightarrow E$ be the inclusion of the chosen fiber. \b[The <long exact sequence of homotopy groups of a fibration> is]
$$
\boxed{\cdots\longrightarrow\pi_{q+1}(B,b_0)
\xrightarrow{\partial}\pi_q(F,e_0)
\xrightarrow{i_*}\pi_q(E,e_0)
\xrightarrow{p_*}\pi_q(B,b_0)
\xrightarrow{\partial}\pi_{q-1}(F,e_0)
\longrightarrow\cdots.}
$$
Its low-dimensional end is
$$
\pi_1(F)\xrightarrow{i_*}\pi_1(E)\xrightarrow{p_*}\pi_1(B)
\xrightarrow{\partial}\pi_0(F)\xrightarrow{i_*}\pi_0(E)
\longrightarrow\pi_0(B)=\{*\}.
$$
The maps $i_*$ and $p_*$ are induced by inclusion and projection on based maps; at degree zero they send a <path component> to its containing or image component. The last part is an exact sequence of <pointed sets>, not in general a sequence of groups.
For $\partial:\pi_q(B)\to\pi_{q-1}(F)$, represent a class by a based cube $a:I^{q-1}\times I\to B$, constant at $b_0$ on its boundary. Lift it from the constant map $e_0$ on the bottom and side faces, using relative disk lifting. Its top face lands in $F$ and is constant on that face's boundary; its class is $\partial[a]$. The choice of lifts does not change the resulting class. For $q=1$, lift a based loop starting at $e_0$ and take the <path component> of its endpoint in $F$. This fixes the boundary-map convention and defines every map in the displayed sequence.
For the splitting assertion, first take $n\geq1$. We use the <cohomological Serre spectral sequence> with coefficient group $G$:
$$
E_2^{s,t}=H^s(B;H^t(F;G))
\Longrightarrow H^{s+t}(E;G),\qquad
d_r:E_r^{s,t}\to E_r^{s+r,t-r+1}.
$$
Since $B$ is <simply connected>, these coefficient systems are constant. The <Eilenberg–MacLane space> $F=K(G,n)$ has $H^t(F;G)=0$ for $0<t<n$, and
$$
H^n(F;G)\cong\operatorname{Hom}(G,G).
$$
For $n=1$ this uses $H_1(F)=G$, with $G$ abelian; for $n>1$ it follows from the <Hurewicz theorem> and the <universal coefficient theorem for cohomology>. Let $u$ correspond to $\operatorname{id}_G$, the <universal cohomology class of an Eilenberg–MacLane space>.
In bidegree $(0,n)$ there are no incoming differentials. For $2\leq r\leq n$, an outgoing differential lands in a vanishing fiber-cohomology row. The only remaining possible differential is
$$
d_{n+1}u\in H^{n+1}(B;G)=0.
$$
Thus $u$ survives. The edge map, which is restriction to $F$, supplies a class
$$
c\in H^n(E;G),\qquad i^*c=u.
$$
By <representability of cohomology by Eilenberg–MacLane spaces>, choose a based map $q:E\to K(G,n)$ representing $c$. On $F$, $q$ induces the identity on $\pi_n=G$ and is an isomorphism on every <homotopy group>, since the other positive groups vanish.
Consider $h=(p,q):E\to B\times K(G,n)$. This is a map of fibrations over $B$, whose map on the fiber is the weak equivalence just obtained. The map on the base is the identity. Comparing the two instances of the <long exact sequence of homotopy groups of a fibration> and applying exactness, or the <Five lemma> in the group-valued range, shows that $h$ induces all <homotopy> isomorphisms. The spaces are path-connected because the base and fiber are. Hence \b[the <splitting of a simply connected Eilenberg–MacLane fibration> gives]
$$
\boxed{E\xrightarrow{\ \simeq_{\mathrm w}\ }B\times K(G,n).}
$$
The crucial use of $H^{n+1}(B;G)=0$ is to lift the fiber's identity class; existence of a section alone was not assumed to prove a product splitting.
If $K(G,0)$ is allowed, its components have trivial positive <homotopy groups>. The <simply connected> base has no monodromy on this discrete set. Obstruction theory on the CW base gives a section in each fiber component, since all positive-dimensional fiber obstructions vanish and the component system is constant. Together these sections give $B\times G\to E$, a weak equivalence on each component. Thus the same conclusion also covers that interpretation.
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