Let $K_{A}, K_{B}, K_{C}$ be the areas of triangle $B P C$, $C P A$, $A P B$, respectively. Let $A'$ and $P'$ be the projections of $A$ and $P$ on side $B C$. Triangles $A L A'$ and $P L P'$ are similar, hence we have
$$\frac{A L}{P L}=\frac{A A'}{P P'}=\frac{A A' \cdot B C}{P P' \cdot B C}=\frac{K}{K_{A}}, \tag{1}$$
where $K$ is area of triangle $A B C$. From (1) we get
$$\frac{A P}{P L}=\frac{K_{B}+K_{C}}{K_{A}}$$

In analogous way we obtain
$$\frac{B P}{P M}=\frac{K_{C}+K_{A}}{K_{B}}, \quad \frac{C P}{P N}=\frac{K_{A}+K_{B}}{K_{C}} . \tag{2}$$
The inequality is equivalent to
$$\frac{A P}{P L} \cdot \frac{B P}{P M} \cdot \frac{C P}{P N} \geq 8$$
that is
$$\left(K_{A}+K_{B}\right)\left(K_{B}+K_{C}\right)\left(K_{C}+K_{A}\right) \geq 8 K_{A} K_{B} K_{C} . \tag{3}$$
Inequality (3) follows by applying three times the inequality $x+y \geq 2 \sqrt{x y}$, where $x, y>0$.

We have equality if and only if $K_{A}=K_{B}=K_{C}=\frac{1}{3} K$, hence if and only if $P=G$, the centroid of triangle $A B C$.

We will use so-called Van Aubel relation, that is
$$\frac{A P}{P L}=\frac{A N}{N B}+\frac{A M}{M C} . \tag{1}$$

In order to prove (1), we use Menelaos Theorem for triangle $A B L$ and collinear points $C, P, N$, and for triangle $A C L$ and collinear points $B, P, M$. We get
$$\frac{C L}{C B} \cdot \frac{N B}{N A} \cdot \frac{P A}{P L}=1 \text{ and } \frac{B L}{B C} \cdot \frac{M C}{M A} \cdot \frac{P A}{P L}=1 . \tag{2}$$
From (2) we obtain
$$\frac{C L}{C B} \cdot \frac{A P}{P L}+\frac{B L}{B C} \cdot \frac{A P}{P L}=\frac{A N}{N B}+\frac{A M}{M C},$$
hence relation (1) since $C L+B L=B C$.
Writing the similar relations to (1) for Cevians $B M$ and $C N$, we have
$$\frac{B P}{P M}=\frac{B L}{L C}+\frac{B N}{N A} \text{ and } \frac{C P}{P N}=\frac{C L}{L B}+\frac{C M}{M A} . \tag{3}$$
It follows
$$\frac{A P}{P L} \cdot \frac{B P}{P M} \cdot \frac{C P}{P N}=\left(\frac{A N}{N B}+\frac{A M}{M C}\right)\left(\frac{B L}{L C}+\frac{B N}{N A}\right)\left(\frac{C L}{L B}+\frac{C M}{M A}\right)$$
and the inequality follows from AM-GM inequality for two numbers.