B) started
"When they began to play it started to rain." Past tense 'started' is correct as the sentence describes a past action using simple past tense.
Both verbs in the sentence refer to past actions ('began' is simple past), so the blank must also be in the simple past tense — 'started'. Option A ('start') is present tense, C ('starts') is present tense, and D ('having started') is a participle form, none of which match the tense sequence required here.