When mathematicians speak of pure mathematics having “intrinsic value” or being for the sake of improving human understanding, what they really mean is that no applied value (in engineering or science usually) is necessarily known at present, but it improves our understanding of mathematics as a system in itself. Why mathematics is this way is not an easy answer, you would have to first have an agreed upon definition of what mathematics actually is, and we don’t have this. Most people have an intuitive notion and “know it when they see it,” but mathematicians and philosophers of mathematics don’t have a clear answer, and different schools like the (neo-)platonists, formalists, constructivists, etc. have different contradictory approaches in even trying to get to a clear definition.
Which gets closer to my real point, mathematics is inherently philosophical and I’d argue it is a kind of analytic philosophy. A close reading of late 19th/early 20th-century history of mathematics, I believe, shows us how closely mathematics developed alongside analytic philosophy, such that at the foundational level at least, they seem to be the same branch of knowledge. It wasn’t too long ago that in some parts of the world mathematics was (rightfully in my opinion) considered a branch of philosophy and there was no clear separation between a philosopher and a mathematician. So, in my mind, mathematics is the way it is, or is beneficial to humans because, or is worth pursuing ourselves instead of handing most of it to machines because of the same reason as philosophy is all of these things: human beings are naturally curious and we satisfy our curiosity by trying to learn and understand things, it’s a natural drive and it’s among the things which help us grow as individuals and as a collective.
Personally, I consider the above a very practical value, so I’d argue that pure mathematics does have “applied” value at a more basic level than what engineering or science disciplines are usually concerned with.