What does it mean that space-time is emergent?

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).