Why B? And why not the others?

Correct answer: B. The rain stopped, and the sun came out.

Both 'the rain stopped' and 'the sun came out' are complete, independent clauses that could each stand alone as their own sentence, so joining two of them properly requires more than just a comma. Pairing a comma with a coordinating conjunction like 'and' is one correct way to link them, which is exactly what the second sentence does. Joining them with only a comma and no conjunction, as the first sentence does, is called a comma splice — a comma alone is too weak a joint for two complete clauses. Leaving out all punctuation, as the third sentence does, is a run-on sentence, an even weaker joint than a comma splice. The fourth sentence tries to combine two separate fixes at once — a semicolon (which can join independent clauses on its own) followed by the conjunction 'and' — but a semicolon and a coordinating conjunction are two different tools for the same job, and standard punctuation uses only one of them, not both together.

Source: BBC Bitesize GCSE English Language: Sentence punctuation and comma splices