A key concept is that gauge fields are associated with constraints. Very roughly, you can think of an emergent gauge field as something that is introduced into the fine-resolution theory as a lagrange multiplier to enforce a constraint, and then at lower resolution it acts like one of the dynamical fields.
This is possible because we can get the lower-resolution effective theory by "integrating out" (or solving the equations of motion for) the finer-resolution degrees of freedom, and since they're all connected to each other through the equations of motion, the effective lower-resolution degrees of freedom get contributions from all of them. So what started as a non-dynamic lagrange multiplier can turn into an effective dynamical field at low resolution.
In general, gauge fields (emergent or not) are associated with redundancies in how the model describes physics. If all of a model's observables are invariant under a given transformation, but the fields are not, then we call that transformation a gauge transformation. Roughly speaking, a gauge field is a field whose presence allows us to have a continuous group of gauge transformations. (We can have a discrete group of gauge transformations without any gauge field.)
Gauge field are also associated with constraints — relationships between observables at the same time. When we use the action principle to derive the equations of motion for a gauge field, we get equations that relate observables to each other at the same time as well as relating observables to each other at different times. Sometimes we distinguish the first type of equation and call them "constraints," since the name "equation of motion" suggests relationships across time. The reason we get constraints this way is related to the fact that the gauge field is partly redundant; it's not invariant under gauge transformations.
I don't know much about Topological Order in condensed matter, so I won't try to say anything specifically about that. Hopefully somebody else will step in and give you a more complete answer.
Section 5 in the paper Wormholes, Emergent Gauge Fields, and the Weak Gravity Conjecture assumes familiarity with field theory, but it's a good reference to add to your list of things to study later (or skim now) because it has a nice review of a relatively simple example of an emergent gauge field.
The paper Symmetry and Emergence puts the idea of emergent gauge fields into a larger context.
