The First Homology Group is the Abelianization of the Fundamental Group.

I follow the proof by Hatcher that the OP outlines.

The first important point is the following, once you have proved that $f = \Sigma_{i,j}(-1)^jn_i\tau_{i,j}$ for your 1-cycle $f$, you need to remember that the group of 1-chains (which contains the 1-cycles as a subgroup) is the free $\mathbb{Z}$-module (ie the free abelian group) on the set of continuous maps from $\Delta_1$ to $X$. Hence, in particular, every element $f$ of this group as a unique expression of the form $n_1f_1+\ldots +n_pf_p$ with $n_i\in\mathbb{Z}$ and $f_i$ a continuous map from $\Delta_1$ to $X$. So, from the equality $f = \Sigma_{i,j}(-1)^jn_i\tau_{i,j}$ you conclude, as noted by Hatcher (third paragraph from the end), that $f$ is one of the $\tau_{i,j}$'s and the remaining $\tau_{i,j}$'s form canceling pairs. This allows you to state this equality between homotopy classes $[\Sigma_{i,j}(-1)^jn_i\tau_{i,j}] = \Sigma_{i,j}(-1)^jn_i[\tau_{i,j}]$. But $\Sigma_{i,j}(-1)^jn_i[\tau_{i,j}] = \Sigma_in_i[\partial\sigma_i]$.

The second important point consists in noting the fact that, $\sigma_i$ being a singular 2-simplex with boundary given by $\tau_{i0} -\tau_{i1} +\tau_{i2}$, you can continuously deform this boundary, through $\sigma_i$, into the constant loop at $x_0$. Hence, one has the following equalities between homotopy classes $[\partial \sigma_i] = [\tau_{i0}]-[\tau_{i1}]+[\tau_{i2}]=0$.

Now, third point, you need to realize this last equality means $[\tau_{i2}] = -([\tau_{i0}]-[\tau_{i1}])$, or in multiplicative notation as you wish $[\tau_{i2}]=([\tau_{i0}][\tau_{i1}]^{-1})^{-1}$. Thus $[\partial\sigma_i]=[\tau_{i0}] -[\tau_{i1}] +[\tau_{i2}]$, being an element followed by its inverse, belongs to the commutator subgroup and so $\Sigma_in_i[\partial\sigma_i]=[f]$ does, meaning that $[f]$ is trivial in $\pi(X,x_0)_{ab}$.

I do not understand a lot of the last paragraph either, but I will give a slightly different proof of the statement based on Bredon, Topology and Geometry, p. 174.

For every point $x\in X$, fix a path $\lambda_x$ from $x_0$ to $x$. Let $\sigma$ be a singular 1-simplex in $X$, i.e. a continuous path $\sigma\colon[0,1]\to X$. Then $\Psi(\sigma):=\overline{\lambda_{\sigma(1)}}\cdot\sigma\cdot\lambda_{\sigma(0)}$ is a loop based at $x_0$, where the dot is juxtaposition of paths and $\overline\lambda(t)=\lambda(1-t)$ is the inverse path. I claim that this induces a well-defined map $\Psi_*\colon H_1(X)\to\pi_1(X,x_0)^{\text{ab}}$, such that $\Psi_*h\colon\pi_1(X,x_0)\to\pi_1(X,x_0)^{\text{ab}}$ is the canonical projection. From this it follows that $h$ induces a monomorphism $\pi_1(X,x_0)^{\text{ab}}\to H_1(X)$.

First well-definedness: Since $\pi_1(X,x_0)^{\text{ab}}$ is abelian, $\Psi$ extends to a homomorphism $\Psi\colon\Delta_1(X)\to\pi_1(X,x_0)^{\text{ab}}$ from the group of singular 1-chains. It is now straightforward to check that boundaries of singular 2-simplices go to zero in $\pi_1(X,x_0)^{\text{ab}}$ (I can elaborate on this if you like), so $\Psi$ induces a homomorphism $\Psi_*\colon H_1(X)\to\pi_1(X,x_0)^{\text{ab}}$ as claimed.

Now if $\sigma$ is a loop based on $x_0$ then $\Psi(h(\sigma))=\overline{\lambda_{x_0}}\cdot\sigma\cdot\lambda_{x_0}$, and the class of this in $\pi_1(X,x_0)^{\text{ab}}$ equals the class of $\sigma\cdot\overline{\lambda_{x_0}}\cdot\lambda_{x_0}\simeq\sigma$, so that $\Psi_*h$ is the canonical projection.