Now let me attempt to give an answer myself.
There are very concrete descriptions of the fibers $Fl_{\epsilon}$ in $Fl_{\mathbb{A}^1}$ for each $\epsilon \in \mathbb{A}^1$.
$Fl_{\epsilon} \cong LG/I_{\epsilon}$, where $LG = G(k((t)))$ is the loop group of the algebraic group $G$, and $I_{\epsilon}$ is the pre-image of the Borel subgroup $B$ under the map $G(k((t))) \rightarrow G$ by evaluating at $t = \epsilon$.
There is also a lattice picture of $Fl_{\epsilon}$ for type A. $Fl_{\epsilon}$ is the moduli space of the following data: a lattice $L$ and a flag $f$ in the vector space $L/(t - \epsilon)L$. When $\epsilon = 0$, we recover the usual lattice picture of the affine flag variety $Fl$.
In Gaitsgory's example above, $G = GL_2$, $Y_0$ is the moduli space of lattices $L$ contained in $L^0 = \mathcal{O} \oplus \mathcal{O}$ with $\dim(L^0/L) = 1$. $Y_0$ is isomorphic to $\mathbb{P}^1$ and we are interested in the closure of its image in $Fl$ as $\epsilon \rightarrow 0$.
Let $a_1, a_2$ denote the two $T-$fixed points of $Y_0$ such that $a_2 < a_1$ as lattices. As $\epsilon \rightarrow 0$, the image of $a_1$ and $a_2$ are the points $(a_1, l), (a_2, l')$ in $Fl$, where $l$ and $l'$ are the lines fixed by $B$ in the respective flag varieties.
Another $T-$fixed point in $Fl$ that is in the closure of the image of $Y_0$ is $(a_1, l')$. For the family of lattices $a_1 + \epsilon^2 a_2$, $l'$ is the right flag to pick in $G/B$ $\forall \epsilon \neq 0$.
Overall, the image of $Y_0$ in $Fl$ as $\epsilon \rightarrow 0$ is two copies of $\mathbb{P}^1$ that connects these three $T-$fixed points.