Here author remarks that $Y_{i}$ closure is homeomorphic to $S^{3}-(K \cup N) $ which we can see as $S^{3}-K$ cut along M with two boundary components homeomorphic to M(interior) and we can then think X tilda as gluing of these later copies. Can someone shed light on this remark, ( I don't see the homeomorphism and the equivalent construction of X tilda).
Write $X_K^\infty$ for the infinite cyclic cover of the knot exterior $X_K:=S^3 \setminus \nu (K)$ and $Y:=\bigcup\limits_{i=-\infty}^\infty Y_i$. We must show that $Y \cong X_K^\infty$. Both are covering spaces of $X_K$ and so we use covering space theory: we will get the required isomorphism if the two covers define the same subgroup of $\pi_1(X_K)$.
Here, recall that the subgroup of $\pi_1(B)$ determined by a covering $p \colon E \to B$ is $p_*(\pi_1(E))$. This subgroup equals the set of loops in $B$ that lifts to loops in $E$ (I am avoiding mentioning base points, but there are fixed). Recall from covering space theory that assigning to a covering $E \to B$ the subgroup $p_*(\pi_1(E))$ gives a bijection between covering spaces of $B$ and subgroups of $\pi_1(B)$ (here I am being loose on the precise statement, but all of this is in Hatcher). For short, let me refer to $p_*(\pi_1(E))$ as ``the group of the cover $p \colon E\to B$".
So we have reduced the problem to showing that the loops in $X_K$ that lifts to $X_K^\infty$ coincide with the loops in $X_K$ that lift to $Y$. Let's first look at the group of $X_K^\infty \to X_K$ and then at the group of $Y \to X_K$
Let's recall the definition of $X_K^\infty$. This is the cover of $X_K$ corresponding to the subgroup $\ker(\operatorname{ab} \colon \pi_1(X_K) \to H_1(X_K) \cong \mathbb{Z})$. Therefore the group of this cover equals $\ker(\operatorname{ab})$, i.e. it consists of those loops $\gamma \subset X_K$ with $\ell k(\gamma,K)=0$, i.e. those $\gamma$ such that $M \cdot \gamma=0$, where $M$ is your Seifert surface.
But now if you look at $Y=\bigcup\limits_{i=-\infty}^\infty Y_i$, you see that is also the condition for a loop $\gamma \subset X_K$ to lift a loop in $Y$: if the lifted curve starts going to one side or another of $Y$, it will cross a lift of $M$ and if you want a loop, it had better come back; in total you must therefore have $M \cdot \gamma=0$.