The key is to start by analyzing the product $$f(x) =\prod_{n=1}^{\infty}(1+x^n)$$ You should observe that each factor has a different power of $x$ and hence the product $f(x) $ above acts as a generating function partitions of a number with unequal parts. Thus we have $$f(x) =1 +\sum_{n=1}^{\infty} p_{d} (n) x^n$$ where $p_{d} (n) $ denotes the number of partitions of $n$ with unequal parts.
Consider next the product $$g(x) =\prod_{n=1}^{\infty}(1-x^n)$$ This is very similar to the product $f(x) $ but due to the negative sign involved in powers of $x$ there is a slight complication. Consider the number $10$ and one of its partitions with unequal parts $(1,9)$. The term corresponding to it in $g(x) $ is $$(-x) (-x^9)=+x^{10}$$ Another partition with unequal parts is $(1,2,7)$ and term corresponding to it in $g(x) $ is $$(-x) (-x^2)(-x^7)=-x^{10}$$ Thus the partition with unequal parts and even number of parts leads to coefficient $+1$ and the partition with unequal parts and off numbers of parts leads to coefficient $-1$. It follows that the coefficient of $x^n$ in $g(x) $ is given by $$p_{e} (n) - p_{o} (n) $$ where $p_{e} (n) $ denotes the number of partitions of $n$ with unequal parts and even number of parts and $p_{o} (n) $ denotes the number of partitions of $n$ with unequal parts and odd number of parts.
Thus we have $$g(x) =1+\sum_{n=1}^{\infty} (p_{e} (n) - p_{o} (n)) x^n$$ and clearly $$f(x) = 1+\sum_{n=1}^{\infty} (p_{e} (n) +p_{o} (n)) x^n$$ as we obviously have $$p_{d} (n) =p_{e} (n) +p_{o} (n) $$ The fact that we only take into account the partitions with unequal parts here is primarily because each factor in both $f(x), g(x) $ uses different power of $x$.
You should also read about Franklin's proof where it is shown via a combinatorial argument that $$p_{e} (n) = p_{o} (n) $$ unless $n$ is of the form $$n=\frac{j(3j\pm 1)}{2}$$ and in this case $$p_{e} (n) - p_{o} (n) =(-1)^j$$
