The general idea is to deliberately shape the geometry of spacetime to allow for effects such as faster-than-light travel (relative to another region of spacetime).
Einstein's field equations link geometry to matter and energy:
G_μν = 8πG/c⁴ · T_μν
Left side describes how spacetime curves (the metric tensor). Right side represents what's causing the curvature (the stress-energy tensor): mass, energy, momentum. The usual approach is: you specify T_μν (what matter is where) and solve for the resulting curvature. Metric engineering takes the backward approach. You pick the geometry you want first: e.g a bubble that carries its contents faster than light relative to the outside universe. Then you work out what T_μν would have to be to produce that geometry: what distribution of energy and stress the equations demand.
The classic example of spacetime metric engineering is the Alcubierre drive.
The limitation with this is that the required T_μν has negative energy density in some regions (the bubble walls) which violate some assumptions about how "reasonable" matter behaves. But Quantum Field theory allows such local regions of negative densities and we have created such regions in labs experimentally already (Casimir effect). We "just" don't know how to sustain the effect at bigger (atomic) distances.