Properties of Gomory-Hu trees
The book states the following definition of a cut:

a cut is defined by a partition of $V$ into two sets say $V^{l}$  and $V \setminus V^{l}$ and consists of all edges that have one endpoint in each partition.

Therefore, removing a cut from the graph should mean that we are removing the edges that go across the cut.
Also, consider the following definition of the &quot;cut associated with an edge $(u,v) \in T$ in $G$&quot;.

Each edge $(u,v)$ in $T$ denotes a partition of $V$ given by the two connected components obtained by removing $(u,v)$ from $T$. Consider the cut defined in $G$ by this partition. We will say that this is the cut associated with $(u,v)$ in $G$.

Now, let us consider your example of Gomory-Hu tree for $K_{4}$. We want to find the cut associated with the edge $(3,0) \in T$. If we remove $(3,0)$ from $T$, we obtain the partitioning $\{1,2,3\}$ and $\{0\}$. This defines the cut in $G$.
Therefore, you should be removing the edges of $G$ that goes across this cut i.e., $(0,1)$ $(0,2)$ and $(0,3)$.
Similarly for the edge $(3,1) \in T$, you should be removing the edges: $(1,0)$, $(1,2)$, and $(1,3)$.
And, for the edge $(3,2) \in T$, you should be removing the edges: $(2,0)$, $(2,1)$, and $(2,3)$.
It means we are removing all edges from the graph. Therefore, the number of connected components in the remaning graph would be $4 = (n + 1)$, for $n = 3$.