Why is $\pi_1(F_g)^{ab} = \Bbb Z \langle a_1, b_1, \ldots , a_g, b_g \rangle$?
Note that $[G, G]$ consists of all products of commutators of elements of $G$. In particular, $[a_1, b_1]\dots[a_g, b_g] \in [\pi_1(F_g), \pi_1(F_g)]$, so the relation $[a_1, b_1]\dots[a_g, b_g] = e$ becomes $e = e$ in the quotient group. This may lead you to believe that $\pi_1(F_g)^{ab}$ has presentation $\langle a_1, b_1, \dots, a_g, b_g\rangle$ and hence must be the free group on $2g$ generators, but that is false. For each $i$ and $j$, all of the elements $[a_i, a_j], [b_i, b_j], [a_i, b_j]$ also belong to $[\pi_1(F_g), \pi_1(F_g)]$, and hence become trivial in the quotient. Therefore
$$\pi_1(F_g)^{ab} = \langle a_1, b_1, \dots, a_g, b_g \mid [a_i, a_j] = [b_i, b_j] = [a_i, b_j] = e\ \forall\ i, j\rangle = \mathbb{Z}\langle a_1, b_1, \dots, a_g, b_g\rangle.$$
That is, $\pi_1(F_g)^{ab}$ is the free abelian group on $2g$ generators, i.e. $\pi_1(F_g)^{ab} \cong \mathbb{Z}^{2g}$.
Said another way, given a group $G$ with presentation
$$\langle r_1, \dots, r_m \mid s_1 = \dots = s_n = e\rangle,$$ then its abelianisation, $G^{ab}$, has a corresponding presentation
$$\langle r_1, \dots, r_m \mid s_1 = \dots = s_n = e, [r_i, r_j] = e\ \forall\ i, j\rangle.$$
In your case, the original group presentation had only one relation, $[a_1, b_1]\dots[a_g, b_g] = e$. When the commutator relations between the generators are added to the presentation for the abelianisation, the original relation becomes redundant (i.e. $[a_i, b_i] = e$ for all $i$ implies $[a_1, b_1]\dots[a_g, b_g] = e$). Therefore, you obtain the presentation for $\pi_1(F_g)^{ab}$ I wrote above, and hence conclude that $\pi_1(F_g)^{ab} \cong \mathbb{Z}^{2g}$.
Added Later: Judging from your comments, you seem to be misunderstanding the notation; for the sake of simplicity, I will only consider a finite set of generators $R = \{r_1, \dots, r_m\}$.
One can form the free group on $R$ which can be denoted simply by $F_R$ or $\langle r_1, \dots, r_m\rangle$. The elements of this group are finite strings in the generators and their inverses which are reduced (i.e. they do not contain products of the form $r_ir_i^{-1}$ or $r_i^{-1}r_i$). The group operation on such strings is concatenation (write one string after the other, then reduce).
Alternatively, one can form the free abelian group on $R$ which can be written simply as $\mathbb{Z}^{(R)}$, $\langle r_1, \dots, r_m \mid\ [r_i, r_j] = e\ \forall\ i, j\rangle$, or $\mathbb{Z}\langle r_1, \dots, r_m\rangle$. As before, the elements are reduced strings, but now, two strings which are rearrangements of each other are considered the same, e.g. $r_4r_1r_2^{-1}r_4$ is the same string as $r_1r_2^{-1}r_4^2$. As we can always reorder our strings, we can write every element in a unique way as $r_1^{k_1}r_2^{k_2}\dots r_m^{k_m}$ where $k_1, \dots, k_m \in \mathbb{Z}$. Then $\mathbb{Z}\langle r_1, \dots, r_m\rangle \cong \mathbb{Z}^m$ where the isomorphism is given by $r_1^{k_1}r_2^{k_2}\dots r_m^{k_m} \mapsto (k_1, k_2, \dots k_m)$. When $m = 1$, this is the map that shows that any infinite cyclic group is isomorphic to $\mathbb{Z}$.
In the comments you were unsure how I went from
$$\langle a_1, b_1, \dots, a_g, b_g \mid [a_i, a_j] = [b_i, b_j] = [a_i, b_j] = e\ \forall\ i, j\rangle$$
to
$$\mathbb{Z}\langle a_1, b_1, \dots, a_g, b_g\rangle.$$
As I've outlined above, they are merely two different ways of describing the same group: the free abelian group on the generators $\{a_1, b_1, \dots, a_g, b_g\}$.