I'll use the analogy of the Lute Sutta.
You hear beautiful music coming out of a lute, a stringed musical instrument. If you break it down into its constituent parts, you cannot find music anywhere.
Similarly, reification (papanca) creates the idea of the self, and also the idea of all the non-self things and their relationship to the self. When the existence of the self i.e. the mental idea of the self ("I am the thinker" - from Snp 4.14) ceases, then suffering ceases.
The music in the analogy is the self. The lute in the analogy is the being or satta (Vajira Sutta), an assemblage of the five aggregates.
The different parts of the lute are like the five aggregates of form, feeling, perception, mental formation and consciousness. Similar to how different parts of the lute work together to make music, the five aggregates work together according to dependent origination to build up and keep functioning the existence of the self, which is just a mental idea.
By cultivating wisdom, one eventually attains Nirvana, uprooting ignorance. Dependent origination is like a row of cards, the first of which is ignorance. When ignorance falls, the five clinging aggregates (SN 22.48) cease, hence the tainted version of mental formation, consciousness, name-and-form, six sense bases, contact and feeling, all fall. However, the (non-clinging) aggregates remain, which is what is meant by the Nibbana element with fuel remaining in Iti 44. With this, craving, clinging, becoming / existence, birth (the idea of a self) and suffering (aging, death and this entire mass of suffering) all fall and cease.
So, in this case, the lute remains, but no music comes out of it anymore. It may make sounds (i.e. the operation of the five aggregates), but it does not make sweet music (the self) any more. The arahat can feel the physical sensations of pain and pleasure, but does not suffer from them. The arahat can also have intentions and plans, without being tainted with defilements and the three poisons.
When the arahat's physical body dies, it's referred to as Nibbana element without fuel remaining in Iti 44. Here, the lute is broken into pieces and no sound comes out of it anymore. The five aggregates are no longer experienced. Also, the appearance of a being (satta), the assemblage of the five aggregates ceases. This is an emergent phenomena.
Back to the original analogy of the lute - when you break down the lute into its constituent parts, you cannot find music. There is no single thing or object such as music. It is simply an emergent phenomena that appears when different parts of the lute work together.
Upon Nirvana, what disappears is the emergent phenomena of a self - a mental idea, and also the rest of the reification of non-self mental ideas that relate ideas of the world to the self. Nothing concrete gets destroyed. No single object or thing that can be isolated gets destroyed.
Upon the physical death of the arahat, what disappears is the emergent phenomena of a being (satta), the assemblage of the five aggregates.
What is an emergent phenomena? If you arrange pebbles on a beach to look like a human face, then from the sky you can see the emergent phenomena, which is the form of a human face. But if you look closely, it's just an arrangement of pebbles.
To say that there is a concrete object called music or in this case the self - is eternalism. To say that there is no music at all or in this case, no self at all - is annihilationism. Instead, we explain the emergent phenomena of music by way of how it emerges from the inter-working of the different parts of the lute. Similarly, we explain the emergent phenomena of the self by the inter-working of the five aggregates, by means of dependent origination. This is discussed in the Acela Sutta.
Just like music, the lute is also an emergent phenomena - it is composed of various parts. Similarly, the self and the being are also emergent phenomena.
The self understood as an emergent phenomena, is the concept of anatta (not-self). The self and being understood as emergent phenomena that is objectified and classified by the mind into concrete things, is the concept of papanca (reification).
"Turning something into nothing" combines eternalism (something) and annihilationism (nothing). Instead, the explanation above takes the middle way of dependent origination and emergent phenomena.
Another analogy similar to the lute analogy is that of the scent of a flower in Khemaka Sutta.
A similar discussion occurs in the Yamaka Sutta on whether the Buddha ceases to exist after death. There it is proven that Buddha as a person is also an emergent phenomena that is not something concrete or something that can be isolated as a single object or thing.
Addendum based on discussions in the comments:
As answered by Andrei, "existence" in "cessation of existence" refers to bhava. Sometimes also translated as "becoming".
This is on the chain of dependent origination after clinging and before birth (the mental idea of the self). Cessation of existence is really cessation of the mental idea of the self, which is a reification, along with the reification that objectifies and classifies things into non-self objects according to their relationship to the self.
It's the cessation of the mental ideas that lead to suffering, which is next on the chain of dependent origination after birth. "Cessation of existence" is not cessation of the non-clinging aggregates. Of course, when ignorance ceased, the clinging aggregates ceased.
For e.g. a piece of cooked meat. To a non-vegetarian, it's delicious food. To a vegan, it's a disgusting thing. If the vegan becomes a arahat, he drops thinking "I am a vegan", "this is my car" and he sees cooked meat as what it really is, rather than associating with his self-identity.
These mental ideas are reification of emergent phenomena, that leads to suffering. The cause of it leads back down the chain of dependent origination all the way to ignorance.