@byyy_t is correct about the meaning of that sentence.
However, it is possible that the original speaker made a small mistake and so it said the opposite of what they actually meant to say. This would confuse the meaning.
So I will explain both ways, and how the meaning can change.
The example sentence basically means this: "they would have been less inclined [to do something] were it not for [a certain person] who changed their opinion, so now they are more inclined to do that thing."
So they did not want to get engaged. But then someone changed their mind. So now they want to get engaged.
But this is phrased in a clever way, so it implies that the person who changed their mind about getting engaged...is the same person who they wanted to be engaged to. So they fell in love with that certain person, so that made them decide to get engaged to that certain person. So he is talking about his male lover, and how being in love made him excited to become engaged to his future husband. This can be a very sweet thing to say about his fiance. Which is why byyy_t commented that way.
But if that does not fit the context, then it is possible that the original source/movie/person made a mistake in their phrasing. Because with one small change in the phrasing, this can make it have the opposite meaning.
So if he does not get along with that certain man, and they are not getting engaged together, and they don't even like each other...then it is possible they said it this way by mistake and that they actually meant the opposite meaning. So what they should have said is: "he would have been MORE inclined to be engaged were it not for the presence of a certain man". So he wanted to be engaged, but this certain man is stopping him from getting engaged. This certain man is in the way of being engaged to someone else that they really like.
This may seem like a strange mistake to make, especially if the source person is a native English speaker.
But sometimes even native speakers will accidentally get their phrasing incorrect, especially when the sentence has more than one negative word or phrase in it. Because using several negatives in the same sentence can become very tricky and change the meaning. So maybe they are trying to repeat the meaning to give it emphasis... but if you try to use too many negatives in the same sentence, then English will cancel some of them out and create the opposite meaning instead. So that might be what happened here. Even native speakers can sometimes make this mistake, if they are not being careful enough.
And in this sentence, both the "less inclined" phrase and the "were it not for" phrase want to negate different parts of the sentence, so it becomes tricky to make the sentence behave correctly.
So this is how it works:
- Using: would be "less" inclined + "were it not for"
That creates this meaning: I am inclined to be engaged because of this certain man. This certain man makes me want to be engaged. He changed my mind so that I now want to be engaged. Without his presence, I would still be less inclined to get engaged, but with his presence that has changed so now I do want to get engaged. He is the reason why I now want to be engaged.
So that uses this phrasing: "my friends told me that he would’ve been LESS inclined to be engaged were it not for the presence of a certain man"
- Using: would be "more" inclined + "were it not for"
That creates this meaning: I am not inclined to be engaged because of this man. This man is stopping me from being engaged. I used to be more inclined to get engaged, but not now. This certain man has changed my mind so now I do not want to get engaged anymore. This certain man is in the way of me getting engaged. This certain man has discouraged me from getting engaged, so now I do not want to be engaged anymore.
So that uses this phrasing: "my friends told me that he would’ve been MORE inclined to be engaged were it not for the presence of a certain man "
Hopefully that helps you find the meaning that makes sense in the context. Especially if the original speaker accidentally used the wrong phrasing for it, which can make this complicated.