There is the citation which is self-sufficient.
"An interval J := (c, d] isa union of countably many disjoint intervals $J_i := (c_i, d_i]$. For each finiten, J is a union of the $J_i$ for $i = 1,..., n$, and finitely many other leftopen, right closed intervals, disjoint from each other and the $J_i$."
End of citation.
What does bold words mean?
Countably many. In the textbook there is no strict definition. However, countable set, accoding to paragraph 1.4, is a set which has cardinality of n or there is 1-1 function from $\mathbb{N}$ onto this set. By default we treat countable set as infinite, i.e. there is 1-1 function from $\mathbb{N}$ onto this set.
Finite. Set is finite if it has n $\in \mathbb{N}$ elements.
So the quetion is: If we divide $J$ into countably many disjoint intervals $J_i$ and then divide these intervals between n intervals and "other" set of intervals, why these "other" intervals have finite cardinality?
For example, we have infinite countable set A. And we substract from A it's finite subset $\bar{A}$: $A\diagdown\bar{A}$. So $A\diagdown\bar{A}$ should be countable infinite, should't it?
I mean that $\infty - n$ is still $\infty$.