I don't understand the red-underlined sentence. What are they saying is contractible? Are they saying $\text{im}|\phi|$ is contractible or $|L| - \{\text{point}\}$ is contractible? And how do either of these being contractible imply that $|\phi|$ is nullhomotopic?
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$|\varphi|$ is not surjective, so it misses at least one point of $|L|$. Call that point “point”. The image is contained in $|L|$ with that point deleted. That space ($L'=L \setminus \left\{\text{point}\right\}$, not $L$ itself) is contractible. Contractible means there is a point $p \in L'$ and homotopy $H:[0,1] \times L' \to L'$, such that $H_1$ is the constant map $L' \to \left\{p\right\}$. The map $\tilde H(t,x) = H(t,|\varphi|(x))$ is a homotopy between $|\varphi|$ and a constant map.
You seem to have some very specific questions like: why is the $m$-skeleton of an $(m+1)$-simplex, with a point deleted, contractible? That may already be an exercise in the book you're reading. Whether or not it is, it might be a good idea to try to prove that, and ask a different question if you run into trouble.
They say that $|L|-$ point is contractible. This is equivalent to saying that there is a continuous map $H:[0,1]\times |L|-point\rightarrow |L|-point$ such that $H(0,x)=x, H(1,x)=p$. You can define $\phi_t(x)=H(t,\phi(x))$ which is an homotopy between a constant map and $\phi$.