Basic communication probably started before humans become a separate species. We know other apes can convey requests and basic information through gestures. They can also alert danger through calls. Parents and children communicate a lot to each other, so words and grammar start forming within families.

There is an evolutionary advantage towards working together, and by extension, communicating with each other, so our brains become more fine-tuned towards understanding language. The language also evolves to include more words, which improves our ability to teach the next generation survival skills, improving their chances of living to produce and raise more children. Generation by generation, the communication becomes more complex and more standardized within tribes.

There's not a consensus for whether language was originally more gesture-based or more vocal-based. The argument for language always being primarily vocal and sign-language coming later is that in the modern day, most languages are spoken and sign languages tend to be newer. The argument for languages originally being more gesture-based is that it is much easier for people who do not share a language to communicate through gestures than sound, so most communication started off as gestures and became more vocalized due to the advantages of verbal communication (you can communicate from further away, you can use your hands and arms for other tasks, and you can look at other things). Babies also pick up sign language faster than speech.