These are basically parentheticals. In utterances (spoken language), if something is inserted which is a meta comment (a comment on a comment) by the speaker on what is being said, this is often set off by em dashes:
Examples:
Em dashes in place of parentheses A pair of em dashes can replace a
pair of parentheses. Dashes are considered less formal than
parentheses; they are also more intrusive. If you want to draw
attention to the parenthetical content, use dashes. If you want to
include the parenthetical content more subtly, use parentheses.
[Bolding mine]
[...]
Upon discovering the errors (all 124 of them), the publisher immediately recalled the books.
Upon discovering the errors—all 124 of them—the publisher
immediately recalled the books.
- I got tired of —let me call it— the monster.
As this is speech, parenthesis are not really a good way to go here.
em dashes
Here is a further definition of this usage:
An em dash—inserted by typing Control+Alt+Minus between the words it separates—signals an abrupt break in thought. It can be
seen as “surprising” the reader with information. If used judiciously
it can mark a longer, more dramatic pause and provide more emphasis
than a comma can.
[...] [Used in writing also]
Example: The actors bowed—except for the horse—knowing they blew
everyone away!
em dash