This では is the copula (だ) followed by the contrastive-は. That is, it's the same では as in これは本ではありません. Since では is normally followed by a negation marker (ない, ありません, etc), you can omit the negation part and still imply the sentence is negative.
This is not a (book)...
(By the way, by avoiding explicitly saying ない, you can often make a sentence sound milder. Instead of clearly saying 寿司は好きではありません, you may want to say 寿司は好きでは…, which is less harsh)
And you can even form a negative question without actually saying ない like this:
Isn't this a book?
(The full sentence is これは本ではない(のです)か？)
Note that this is expecting a positive answer (the speaker is thinking it is a book).
This is what's happening in your sentence, too. お客様がとても退屈しているのだ means "It's that my clients are very bored (because of the long meetings)". Changing のだ into のでは means you are making an implicit negative question: "isn't it that ...?". So a literal translation is like:
He is thinking "isn't it that my clients are very bored?"
Or you can translate it simply like "He thinks his clients may be very bored."
TLDR: This のでは is an (implicit) negative question marker with an explanatory-の ("isn't it that ～?").