I don't understand how the highlighted isomorphism follows. And why is every object in $\mathbf {Set}\times\mathbf{Set}$ is a sum of copies of $(1,\emptyset)$ and $(\emptyset,1)$?
Next, right after the density theorem, there's this example:
By the theorem, the colimit of $H_\bullet \circ P$ is $X$. This example says that the colimit is the sum of five representables, namely $H_K+H_K+H_K+H_L+H_L$. How does this follow from the theorem?
I think $1$ is a one-point set, and "sum" here is coproduct. In the category of sets, coproduct is disjoint union. Likewise in $\textbf{Set}\times\textbf{Set}$ the coproduct of $(A_1,B_1)$ and $(A_2,B_2)$ is $(A_1+ A_2,B_1+ B_2)$. This extends to coproducts of more than two objects, even infinitely many objects.
So $$(\{a,b,c\},\{d,e\})\cong(\{a\},\emptyset)+(\{b\},\emptyset)+(\{c\},\emptyset) +(\emptyset,\{d\})+(\emptyset,\{e\}) \cong(1,\emptyset)+(1,\emptyset)+(1,\emptyset) +(\emptyset,1)+(\emptyset,1)$$ etc.