Jǐn gēn zhe dì yī shēng léi xiǎng, dàyǔ jiù xià le qǐlái.
Right after the first clap of thunder, heavy rain started falling.
Tips
grammar
紧跟着 is 紧跟 (to follow closely) + 着 (aspect particle indicating continuation). It can function as an adverb meaning 'immediately after' or as a verb phrase meaning 'following closely behind'.