This trope is so common in kdramas especially. It's generally used as a contrivance to keep an otherwise unlikely pair together for some reason and allow them some "fake" skinship, while acting as an obstacle and still allowing the possibility of a third (or fourth) party to interfere with our OTP.
The first time I saw this trope was in My Lovely Sam Soon. At the time I watched this drama, this trope seemed fresher than it does now (I have seen it in other genres, but I can't think of any examples). Now that I've seen it so many times, it's lost most of its appeal. Jin Heon proposes the contract to get his mother off his back about going on blind dates. In return, he offers Sam Soon money to cover the mortgage of her house. In this case, the fake relationship gives Jin Heon an excuse to blow off Hee Jin, and the opportunity for Sam Soon to keep Hyun Woo at bay as he tries to win her back.
It also has the added benefit of putting the OTP in situations they wouldn't normally be in together as a boss and employee. The best example of this is the hotel opening Jin Heon drags Sam Soon to, which becomes a major catalyst for the actual pairing of the OTP. It's due to this party that Jin Heon instinctively fights for Sam Soon's honor, opens his heart and his past to her, and gets back together with Hee Jin, helping him realize their past is not the same as their future. It also opens Sam Soon up to confessing her feelings for Jin Heon and exposing her insecurities and exposing her wounded heart.
This use of the RC is also used in Lie to Me. Ki Joon uses the relationship to help him in his business dealings with President Chang, bringing Ah Jung along to exploit the friendship she developed with him and his wife. Ah Jung uses this clause to host the housewarming party, where Ki Joon kisses her for the first time, acknowledging his feelings for her.
The RC can also be a tool used to create opportunities for skinship between the OTP. A common clause, and one used in My Lovely Sam Soon is that the couple may need to have some limited skinship to sell the relationship to others (although it's not used much in My Lovely Sam Soon). In Coffee Prince, Eun Chan agrees to some physical contact—as long as she's paid for it. This leads to their (and her) first kiss, which in turn leads to her developing feelings for Han Kyul.
In Mary Stayed Out All Night, Mary uses this aspect of her contract relationship to not only have skinship with Moo Kyul, but to be able to get her father's approval to stay overnight at his house. The RC with cohabitation trope also comes up in Full House, My Girlfriend is a Gumiho, and even My Girl, even though in the latter the contract is not to be a couple, but cousins. As we all know, living under one roof in kdramaland always leads to the OTP eventually falling for one another, because proximity becomes love.
Another useful aspect of the RC is the barrier it inevitably creates between the couple. Most seem to have the clause that the couple will never, ever have feelings for one another/date for reals. Of course, this adds extra tension to the relationship, and the pair has to overcome this additional obstacle to progress. My favorite way to circumvent this clause is the one Jin Heon uses in My Lovely Sam Soon. How adorable is it when he twists the contract so that Sam Soon and Jin Heon have to date for 100 years, he lends her 10 million dollars, to be repaid over her entire life, and "skinship is allowed."
It's not surprising that writers add RCs to the plots of their shows. It paves the way for so many developments in the relationship while still creating a barrier between the leads. The OTP can have skinship, live together, and have more opportunities to spend time with one another, yet they can't have a real relationship, because it would break their contract. However, even though this trope is useful and comedic, let's find a new way to create these situations for future dramas, okay, writers?