Why is the "actual" gauge group of the standard model $SU(3) \times SU(2) \times U(1) /N$?
Baez actually has another paper (with Huerta) that goes into more detail about this. In particular, Sec. 3.1 is where it's explained, along with some nice examples. The upshot is that the hypercharges of known particles work out just right so that the action of that generator is trivial. Specifically, we have
Left-handed quark Y = even integer + 1/3
Left-handed lepton Y = odd integer
Right-handed quark Y = odd integer + 1/3
Right-handed lepton Y = even integer
Because these are the only values for known fermions in the Standard Model, that generator does nothing. So basically, you can just take the full group modulo the subgroup generated by $(\alpha, \alpha^{-3}, \alpha^2)$ -- where $\alpha$ is a sixth root of unity.
There's also this paper by Saller, which goes into greater detail about the "central correlations" of the Standard Model's gauge group, but in a more technical presentation. Saller also goes into some detail in chapter 6.5.3 of his book.
How can we see that the group $N$ generated by $$ g = (e^{2\pi i/3} I, -I, e^{i\pi /3}) \in SU(3)\times SU(2)\times U(1) $$ acts trivially on all fields in the Standard Model?
First of all, note that $g$ is in the center of $SU(3)\times SU(2)\times U(1)$. Therefore its representative in the adjoint representation is the identity. Since gauge bosons transform in the adjoint representation, $N$ acts trivially on them.
The left-handed lepton fields are in the trivial representation of $SU(3)$ and are an $SU(2)\times U(1)$ doublet, $$\Psi = \begin{pmatrix} \nu_L \\ \psi_L \end{pmatrix}.$$ Baez is using a normalization of charge such that these fields have $U(1)$ charge $-3$., so they also transform trivially under $N$. The right-handed lepton $\psi_R$ has $U(1)$ charge $-6$ in this system, so it too is trivial.
The left (right) handed quarks have $U(1)$ charge $1$ ($4$ or $-2$) in this system and transform under $SU(3)$. It is simple to see that they also transform trivially under $N$.
(Note that various sources put the $1/3$ ratio between quark and lepton charges in various places, so be careful comparing.)
The main point is that if one has a consistent gauge theory containing matter with gauge group $$G:=SU(3)\times SU(2)\times U(1),$$ and if one divides $G$ with a normal subgroup $N$, then the matter representations of the matter fields could potentially become multi-valued. However, it is possible to choose $N=\mathbb{Z}_6$ in such a way that the standard model matter fields are invariant. And $N=\mathbb{Z}_6$ happens to be the maximal possible in this case.
See also this & this Phys.SE posts and links therein for a similar discussion for the electroweak sector.
See also this related Phys.SE post.
References:
J.C. Baez, Calabi-Yau Manifolds and the Standard Model, arXiv:hep-th/0511086.
D. Tong, JHEP 07 (2017) 104, arXiv:1705.01853. (Hat tip: knzhou.)