Explain why the number of integer solutions is equal to the number of integer partitions of $n$

HINT: A partition of $n$ is completely specified when we know the number of parts of each size. For instance, the partition $2+2+2+1+1$ of $8$ is completely specified by saying that it has $3$ parts of $2$ and $2$ parts of $1$. Interpret the numbers $x_1,ldots,x_n$ as numbers of parts in some way. I’ve put a further hint in the spoiler-protected box below.

Added: Here is an example to illustrate more fully the correspondence between solutions in non-negative integers to $$1x_1+2x_2+3x_3+ldots+nx_n$$ and partitions of $n$. Specifically, I’ll match each of the $7$ partitions of $5$ with the corresponding solution to $1x_1+2x_2+3x_3+4x_4+5x_5=5$.

$$begin{align*} &color{red}5=color{blue}5:\ &1cdot0+2cdot 0+3cdot0+4cdot0+color{blue}{5cdot 1}=color{red}5\ &color{blue}{x_5=1};x_1=x_2=x_3=x_4=0\ &\ &color{red}5=color{blue}{4+1}\ &color{blue}{1cdot1}+2cdot 0+3cdot0+color{blue}{4cdot1}+5cdot0=color{red}5\ &color{blue}{x_4=1,x_1=1};x_2=x_3=x_5=0\ &\ &color{red}5=color{blue}{3+2}\ &1cdot0+color{blue}{2cdot1}+color{blue}{3cdot1}+4cdot0+5cdot0=color{red}5\ &color{blue}{x_3=1,x_2=1};x_1=x_4=x_5=0\ &\ &color{red}5=color{blue}{3+1+1}\ &color{blue}{1cdot2}+2cdot0+color{blue}{3cdot1}+4cdot0+5cdot0=color{red}5\ &color{blue}{x_3=1,x_1=2};x_2=x_4=x_5=0\ &\ &color{red}5=color{blue}{2+2+1}\ &color{blue}{1cdot1}+color{blue}{2cdot2}+3cdot0+4cdot0+5cdot0=color{red}5\ &color{blue}{x_2=2,x_1=1};x_3=x_4=x_5=0\ &\ &color{red}5=color{blue}{2+1+1+1}\ &color{blue}{1cdot3}+color{blue}{2cdot1}+3cdot0+4cdot0+5cdot0=color{red}5\ &color{blue}{x_2=1,x_1=3};x_3=x_4=x_5=0\ &\ &color{red}5=color{blue}{1+1+1+1+1}\ &color{blue}{1cdot5}+2cdot0+3cdot0+4cdot0+5cdot0=color{red}5\ &color{blue}{x_1=5};x_2=x_3=x_4=x_5=0 end{align*}$$