What is the purpose of the Mahayana 'emptiness' doctrine?

Further to e.g. this answer, which may be an overview of what Mahayana teaches about emptiness, my question is why does Mahayana teach that?

Well, emptiness is not a Mahayana's invention. They just further elaborated on it. It's been there all along in the Buddha's Teaching. There's a whole Division in the MiddleLength Discourses dedicated to the theme, it's called the Division on Voidness - SunnataVagga, composed of MN 121 thru 130. While it's mainly part of the Insight side of training, but it's also an important theme for the cultivation of Sila and Samadhi:

"Bhikkhus, for direct knowledge of lust, three things are to be developed. What three? Emptiness concentration, markless concentration, and wishless concentration." ~~ AN 3.183 ~~

Short answer: because understanding that phenomena are not "solid", but are in fact an interpretation/imputation, removes craving and passions in regards to them, which prevents dukkha that arises from that.

Long answer:

Normally we think that world is like a fish tank - a 3D environment full of objects, some of which are food, some friends, some dangerous sharks etc. with ourselves being one of the fishes in the pool. Then we spend most of our lives chasing the food and attractive fish, avoiding sharks, and trying to occupy the best spot in aquarium.

We assume that the world actually exists as it appears to us. We take it for granted that objects are objects, that some of these objects are awesome and some are not, that we are in competition with the others, and that the objective of the game is to achieve happiness by arranging a perfect environment for ourselves, and then maintaining it as long as possible.

This entire dream is based on an idea of solidity of the external world, an idea that things are actually out there as they appear, awesome or bad independently of our mind, and that therefore obtaining them or avoiding them is the right thing to do, and that therefore failing at that is the right reason to be upset and to experience suffering.

However, attaining an insight into "how things really are" - which is to say in how things are not solid, and how the mind of illusory solidity develops (process known as D.O.) -- having a direct insight into all this pulls the rug from under the initial illusion of solidity and the entire complex of behavior that comes with it.

The traditional analogy is rope that seems like a snake. Once you see it's a rope, the fear of getting bitten is gone and has no basis for ever coming back.

The next question naturally is, what exactly are we talking about when we say things are not solid, and how does that help remove the craving and other kleshas.

When we say things are not solid, we are saying that each individual thing is in fact multiple independent phenomena that seem to be lumped together, but only when observed from a certain perspective. In fact, this perspective is the main force that gives things their illusory solidity. Or, put in other words, much of what things seem to be actually comes from ourselves, from our own interpretation. Remember the snake-rope.

Once we get a very clear first-hand insight into how we "project" (impute) our perspective onto things, and learn to loosen that up, we realize that much of the importance we assigned to things is in fact our own game of make-believe. We also realize how many conflicts and passions in the world originate from people taking their own perspectives/interpretations all too seriously and assuming things are the way they look to them, and that all other ways of seeing are wrong.

Just like individual phenomena, entire world as it appears in our subjective experience is a giant interpretation that we assemble by looking at things a certain way, and pulling together certain strings of ideas to form a (semi-)coherent picture. And then of course our idea of ourselves, our feeling of "me-ness", is similar construct, made up of a million separate strings we strive to maintain intact. Emptiness of self is studied at length in Pali Canon, under the name of Anatta.

What Buddhists call the circle of Samsara (the cycle of subsequent generations of people inheriting this ignorant way of perception and passing it on to their children) and the struggle that comes with it - entirely depends on ignoring the fact that reality is our interpretation and that our interpretation is based on our limited exposure, and that our exposure comes from the circumstances, and that our circumstances come from results of previous actions, and that previous actions came from taking the illusion of solidity at the face value.

But once we understand how it all works, we can be free from it to a large degree, at least we can certainly be free from the subjective dukkha, and oftentimes even free from getting ourselves into troublesome situations that involve conflict of ideas.

Why? Because when we see that things are our own projection/interpretation/imputation - we no longer are seriously attached to them, we are more flexible, we allow for other interpretations, we see things from multiple perspectives which is a more robust way of seeing things. It turns out that most of our cravings, conflicts, and passions - most of our trouble and suffering - came from the assumption that things are real and actually possess all those qualities and characteristics, and now that we know they are/do not, now we can be free from all that.