What does it mean that space-time is emergent? Edward Witten (Institute for Advanced Study in Princeton), Fields medallist, the discoverer of M-theory and considered a genius by many, recently gave one of his rare interviews (https://www.quantamagazine.org/edward-witten-ponders-the-nature-of-reality-20171128).
In it he states:

I tend to assume that space-time and everything in it are in some
sense emergent.

Could anyone try to shed some light on what is meant by that? How does one have to imagine "emergent space-time"? How does it work to have something and from that build space and time?
 A: 
How does it work to have something and from that build space and time?

It very much depends on the ontologic charge you attribute to something.
Considering with Kant that space and time are a priori attributes of cognition (along with causality), it seems to us humans that no object can be that something since our imagination will automatically endow whatever candidate object with some spatial and temporal structure (there is a discussion of this point in Geometry, pregeometry and beyond (Meschini, 2004)).
So the answer has to be purely mathematical. There will be no intuitive notion of what the basis of emergence for spacetime is. What may be more fruitful is to look for what spacetime is in our current representation in physics, and notably what it provides. If some mathematical structure can provide the same things as spacetime, and can be built from lower-level concepts, then these concepts will count as the something from which spacetime can emerge.
In relativity spacetime (defined as a manifold of events) provides a picture of causality. Some researchers take causality and events as fundamental concepts, and attempt to build spacetime from so-called causal sets of events. See for example A potential foundation for emergent space-time  (Knuth, 2014).
Other researchers see spacetime as a provider of symmetries and algebraic structures to be obeyed by physical systems. They can work on a mathematical exploration of these structures and get spacetime from there. See for example The Structure of Spacetime and Noncommutative Geometry (Lizzi, 2008).
A: There is a recent paper which considers emergence of spacetime geometry from quantum entanglement:
Bulk Entanglement Gravity without a Boundary: Towards Finding Einstein's Equation in Hilbert Space by ChunJun Cao, Sean M. Carroll
Abstract
We consider the emergence from quantum entanglement of spacetime geometry in a bulk region. For certain classes of quantum states in an appropriately factorized Hilbert space, a spatial geometry can be defined by associating areas along codimension-one surfaces with the entanglement entropy between either side. We show how Radon transforms can be used to convert this data into a spatial metric. Under a particular set of assumptions, the time evolution of such a state traces out a four-dimensional spacetime geometry, and we argue using a modified version of Jacobson's "entanglement equilibrium" that the geometry should obey Einstein's equation in the weak-field limit. We also discuss how entanglement equilibrium is related to a generalization of the Ryu-Takayanagi formula in more general settings, and how quantum error correction can help specify the emergence map between the full quantum-gravity Hilbert space and the semiclassical limit of quantum fields propagating on a classical spacetime.
https://arxiv.org/abs/1712.02803
--
Another highly interesting approach is the following by Carlo Rovelli:
"Forget time" by Carlo Rovelli
Abstract
Following a line of research that I have developed for several years, I argue that the best strategy for understanding quantum gravity is to build a picture of the physical world where the notion of time plays no role. I summarize here this point of view, explaining why I think that in a fundamental description of nature we must "forget time", and how this can be done in the classical and in the quantum theory. The idea is to develop a formalism that treats dependent and independent variables on the same footing. In short, I propose to interpret mechanics as a theory of relations between variables, rather than the theory of the evolution of variables in time.
https://arxiv.org/abs/0903.3832
This is also the topic of his new popular science book:
Carlo Rovelli: The order of time, 2018
A: Well one possibility relies on a theorem of Malament that the causal structure of spacetime determines the metric upto a conformal factor. And this factor can be determined from specifying a density. This has inspired Sorkin's causal set theory as one attack on the problem of an a priori quantum gravity. Here, the essential structure is discrete - hence sets - with the spacetime manifold being emergent as a continuum approximation. He characterises his programme as "order+ number = geometry".
Loop quantum gravity on the other hand quantises gravity in a background independent fashion. In this theory there are, in analogy to the quantum operators of position, also quantum operators of area and volume. These have a discrete spectrum and so again discretising spacetime.
A: It is believe that spacetime is emergent.It is not clear, however, where the process of its emergence takes place. The majority of physicist tend to suppose that spacetime is created on the Planck's scale at a distance of trillion.
Time can be viewed as an emergent property: whenever any change occurs anywhere in the universe - in other words whenever the state space of the universe undergoes any change - then Time itself happens and 'notches up' one tick.
