This one was tough for me to work through. However, I THINK it follows these lines:  'les couple(s)' here means 'the relationships'.

« les couples des autres » means 'the relationships of the other(s)/ones' - « les couples (de les → des) autres ». 

An important point here is 'les autres' is the indefinite pronoun, (one of a few that take a definite article), and that «autres» is not an indefinite adjective. With a definite article preceded by preposion de, contraction occurs, hence « de les » contracts.

I hope this is close to the mark but happy to have clarification from others.

https://www.lawlessfrench.com/grammar/definite-article/

"Les autres" and "d'autres" can both be translated into English as the indefinite pronoun "others". The distinction should be that the former means "all others", and the latter "some others". I would have expected it to be "d'autres" in this case, but apparently I would be wrong.

There is the usual problem of deciding whether to use the definite or partitive article when English uses no article. e.g. "J'aime le chocolat", but "Je mange du chocolat" and "J'aime manger du chocolat". Since the character presumably breaks up some other couples rather than all of them, I would have expected "d'autres" here.

I wonder if there is something specific to "autres" going on here? I notice (based on google search results) that people also say "on a besoin des autres" and not "on a besoin d'autres", which is also the opposite of what I expected. The only hits for "on a besoin d'autres" are when it is followed by a noun. Is the problem that a listener would rather expect "d'autres" to be an adjective and would be waiting for a noun to come? 

Without having seen the movie, I am assuming « les (des) autres » is correctly used in the review. If so, I think it is probably meant in the specific that he is trying to break the relationships of the couples in the film; alternatively I guess it could be general - he aims to break up every couple he comes across everywhere (and would break up every couple if possible), but prima facie less likely I think. 

Perhaps, the review's author could clarify ?

No, it's not limited to the specific people in the film, it's his "business".

https://www.imdb.com/title/tt1465487/

Alex and his sister run a business designed to break up relationships. They are hired by a rich man to break up the wedding of his daughter. The only problem is that they only have one week to do so.

It is a group limited to those he is engaged to disrupt. He has some 'clients',  his job is to disrupt all of their relationships. Don't know the answer but if the author has written thinking of a specific group of people, I can see why « des autres » can be used.