How do people acquire and develop language?
The answer depends on which theory of language acquisition you set more weight in. This is because learning, as every other behavior, is something that requires cognitive, behavioral, physical, social, and psychological input.
Noam Chomsky first proposed that there is a linguistic corpus located in our brains, serving as a form of enabler for us to encode and decode language. He called it the LAD, or Language Acquisition Device. (1960) Keep in mind that this is still a hypothesis as most of the studies in L2 acquisition are either hypothetical or theoretical in nature.
While many areas of the brain are associated with the development of language, memory encoding and retention, and language loss (think: aphasia) the most accepted notions of L2 are the following:
Albert Bandura- (1977) Social learning gives us the input that we need from the community and our environment to develop language skills, while mimicking others: parents, peers, everyone.
B.F Skinner, (1938) one of the first proponents of classical and operant conditioning adds on to the L2 debate by offering that neutral, positive, and negative reinforcers are essential in the retention and development of language. Therefore, it is a process that requires feedback, reinforcement, and motivation.
Stephen Krashen (1983), one of the highest authorities in SLA, proposes a monitor language theory (thinking before speaking), natural order hypothesis (all languages are learned in a predictable, formulaic pattern) and an affective filter theory (learning can only occur when the input channels are free from bias, stress, or fear).
As you can see there are plenty of theories and hypotheses related to SLA, and no single one is used as absolute. Start with these essential foundations to SLA and you will be able to see how much recent research has been done, parting from these very frameworks.
People normally acquire their first languages as infants. Several linguists have theorized that language acquisition "starts", in a sense, by our having a built-in or hard-wired capacity for language. Young babies acquire language by imitating their parents. They learn first basic words that refer to common objects or actions. They learn to associate noises with actions by or objects by repeatedly experiencing both in close conjunction. The ability to understand syntactic relationships and abstract concepts occurs later. Humans have the greatest ability to acquire languages before the ages of seven. As we grow older, though, we can continue to learn new languages.