A cool-sweet healing cohabitation romance of Ureong-i, the best housekeeping neat freak, and Seungha, the weary loner in his hectic routine.
Removing all the modifiers in the sentence, you have
우렁이와 승하의 로맨스
So there are two characters 우렁이 and 승하, and there making a romance. 우렁이 means a freshwater snail, and there is a folklore about a snail doing house chores for a farmer. (References in Korean #1 #2)
And let's take a closer look at the modifiers for each of 우렁이, 승하, and 로맨스.
살림 만렙 알뜰살뜰 깔끔쟁이 우렁이
살림 means housekeeping, and 만렙 is something like a maestro or top-level. Compound of Chinese character 만(萬; ten thousand, or figuratively a very big number) and 렙(from English 'level'), it means the highest possible level of a video game.
EDIT: I just noticed that 만 for 만렙 might actually be 만(滿; full, packed), thus "full level, complete level" in a video game. I don't have any academic backup for either etymological theory.
알뜰살뜰 means one is thrifty, sparing, and good at 살림.
깔끔쟁이 refers to a person who is more or less obsessed to keeping things clean and tidy. It may have a derogatory nuance.
Thus, Ureong-i, the best housekeeping neat freak.
바쁜 일상 속에 지치고 외로운 승하
I think this part is plain textbook Korean. Seungha, the weary loner in his hectic routine.
상큼달콤한 힐링 동거 로맨스
상큼달콤한 is a compound of 상큼하다 and 달콤하다. 상큼하다 means cool and refreshing like a lemonade or a fresh strawberry. 달콤하다 means sweet and tasty like chocolate or a lollipop. Both are primarily used to describe flavor or scent, but here it is understandably used to describe a "로맨스(romance)."
힐링 is from English 'healing'. In a contemporary Korean context, it is used to describe a media production (movie, TV series, anime, webtoon, etc.) or to describe an experience (having a cup of coffee at your favorite coffee shop, seeing your friend's lovely pet dog, spending a weekend at a luxury hotel, etc.) that soothes, comforts, and mentally or emotionally "heals" you. I don't think there's an English equivalent to describe this kind of specific meaning, is there?
동거 means cohabitation, with a strong implication of a not married couple, but it doesn't always have to be a sexual relationship (although it is also strongly implied by the word). It can also be used in a situation of two or more non-sexually related friends. (Like those in the Big Bang Theory or whatever.)
로맨스 is from English "romance", and I think shares most of its meaning with the English word. It can refer to a specific emotion, a literary genre dealing with such emotion, or a piece of fiction of such genre.
Thus, a cool-sweet healing cohabitation romance. It's a romantic story, where the two characters live together, and the overall tone will be comfortable and soothing, as opposed to a comedy or a crime thriller.