I'll try to explain this from the Theravada perspective, which I think is more or less the same as Madhyamaka emptiness, once you analyze it deeply.
In addition to this answer, please also see "Linking Madhyamaka emptiness to Theravada emptiness through papanca".
From Sutta Nipata 4.14, we read:
"I ask the kinsman of the Sun, the great seer,
about seclusion & the state of peace.
Seeing in what way is a monk unbound,
clinging to nothing in the world?"
"He should put an entire stop
to the root of objectification-classifications:
'I am the thinker.'
Objectification-classification in this translation means the same as "reification".
Ven. Thanissaro, the translator, commented on this:
On objectification-classifications and their role in leading to
conflict, see Sn 4.11 and the introduction to MN 18. The perception,
"I am the thinker" lies at the root of these classifications in that
it reads into the immediate present a set of distinctions — I/not-I;
being/not-being; thinker/thought; identity/non-identity — that then
can proliferate into mental and physical conflict. The conceit
inherent in this perception thus forms a fetter on the mind. To become
unbound, one must learn to examine these distinctions — which we all
take for granted — to see that they are simply assumptions that are
not inherent in experience, and that we would be better off to be able
to drop them.
In our minds, we have the idea of "I am the thinker" i.e. the idea of the self. That's the primary object in existence in our reality. We also have the idea of non-self objects i.e. everything else. We objectify and classify everything around us, into non-self objects, according to their relationship to the self. For e.g. my hand, my car, not my friend, not my country.
When you look at the waters of the sea from up close in a boat, you may feel fear and insecurity, especially if you don't know how to swim and have motion sickness. To the sailor, it's a source of joy and adventure. To the fisherman, it's a source of livelihood and he sees it like a mine or oil field. To fish deep in the sea that has never left the waters, the concept of water doesn't occur to it at all, as it does not know any other reality.
Another example - a piece of cooked meat appears like delicious food to the meat eater, and it appears repulsive to the vegan. To a honey bee, it appears like dirt because it's not its food.
These examples go to show that objects do not have the meaning given to it by the mind. In fact, some of these are not even objects, except that they have been objectified by the mind.
What's a body of water to me is nothing at all (or perhaps everything) to the fish. The waters of the great sea, as a place to sail and swim, and as a body of liquid, doesn't really exist, except in my mind. It certainly doesn't exist in that way to the fish.
What's delicious food to me, is dirt to the honey bee. So, the delicious food doesn't really exist, except in my mind. The dirt doesn't really exist, except in the honey bee's mind.
This concept is called papanca, which is objectification plus classification, also known as reification. And it's related to anatta (the teaching that all phenomena is not self), because papanca is when non-self things are reified into objects and they are classified relative to the self. The idea of the self is also papanca.
This does not mean that things don't exist, except in my mind. It means that things don't exist as how my mind thinks it does.
Let's say you see images of attractive people on a screen. To an ant, it's just lights of different colors. If a mouse sees them - it's just an image of what it may perceive as humans - a threat. But to you, they are attractive people. So why are they "attractive people"? It's because that's how your mind objectified and classified them relative to yourself. That's how your mind reified them. That's papanca.
Fully enlightened persons like the Buddha and the arahants see things the way they are, without objectifying and classifying them relative to the idea of the self.
So according to Snp 4.14, to put an end to reification, you must first put an end to the root of reification, which is the mental idea of the self, the thought "I am the thinker". And the way to achieve this is to increase wisdom and weaken ignorance. That would weaken craving, clinging, becoming and birth (of the idea of the self). For this, please see this answer on the South Indian Monkey Trap.
And how do you increase wisdom and weaken ignorance? For that, we have the Noble Eightfold Path.