My mother has been asking me to explain why anyone would study grammar and my rhetorical questions (why do you bother to learn the names of all the plants in your garden) weren't cutting it anymore. I've put off writing this because I'm afraid that it will get out of hand, but here goes:
Grammar has an elitist tinge to it. Having "correct grammar" is generally seen as a good, if perhaps snobby, thing. As a result, people who use dialects or languages that don't obviously conform to grammatical norms are often looked down on, their language disparaged or even banned. Instances can be as trivial as radio hosts getting complaint letters about their perceived errors or as grave as deaf children being forbidden to use sign language. Scientific studies of grammar can help us put the lie to the prejudicial beliefs behind such practices. By describing the complicated grammar behind sign languages for the deaf, by showing that expressions such as he be workin' and he workin' are not sloppy variants of he is working but rather two standard (within the dialect) forms that convey distinct meanings, and by demonstrating that singular they has an impeccable pedigree, the study of grammar can open people's eyes to the real problems of racism, elitism and ignorance that are masked by false justifications based on grammar.
Linked to the above is the need to help people learn a second language. This works two ways: immigrants need to learn the dominant language of their new land and the rest of us are bettered by studying other languages. Either way, there is a great deal of evidence showing that, on average, in otherwise comparable situations, learners who study the grammar of a second language acheive higher levels of proficiency than those who don't. Obviously learners need teachers or textbook writers who have a good understanding of grammar.
The law can turn on the placement of a comma or the meaning of a word. Careful study of grammar can help us adjudicate such questions. It can assist us in identifying forged documents and separating real confessions from other more benign utterances.
Artificial intelligence, machine translation, and information retrieval
The cost of translating documents can be stupifying. If we could develop high-quality translation software, the savings would be imense. Natural language search is a problem that has obvious application to everyone who can't find what they're looking for on the internet. Humans want to be able to communicate with machines through natural language rather than buttons switches and typed commands.
In general, most AI is moving away from hand coded rules towards probabilistic heuristics based on massive corpora, but I would say that there is still a need for people working on such systems to have a deep understanding of grammar.
Some people who suffer a brain injury discover that the grammaticality of their speech is affected, and different injuries have different effects. An understanding of grammar can help us to understand the brain.
The grammar of any natural language is a terribly complicated thing. People who like puzzles will often be intrigued by its nuances and will yearn after the chance to discover something that had previously escaped notice. In other words, many people study grammar for fun.
The above is overly simplified and there are many other reasons besides, but hopefully this will be enough to prod my mother and others into considering what they might be.