As far as i understand it:
They hit their material with a laser, it transitioned to another (higher energy) state, and that state decayed again (quickly).
An excitation can be observed e.g. in a hydrogen atom: Hitting it with light of just the right frequency excites it to a higher energy state, i.e. the electron goes into a (energetically) "higher" quantum state and that state decays emitting light.
In the experiment it's not a hydrogen atom, but a many particle system. In such systems, e.g. in crystals, the lowest energy state may be one in which the electrons (and nuclei) obey a macroscopic order stretching the whole crystal, and arrange in a highly coordinated state. One commonly known example of such behavior is ferromagnetism, where due to quantum mechanical (exchange) interaction the valence electrons of a crystal domain preferably have their spin point in the same direction to reach lowest energy state (that way the exclusion principle reduces electric interaction between same charged electrons).
One such many particle state is a charge density wave. There the electrons are not evenly spread across all (equivalent) lattice sites, instead their density may for example alternate between high and low, alternating from layer to layer in a lattice(*). In their material that is the lowest energy state. They managed to get the material to another energy state, where the charge density wave is in a perpendicular direction, and they showed, that that state quickly decays and the crystal goes back to the ground state.
I think the remarkable thing here is, that they observed quantum state transitions for a larger (many particle) system.
As for the "unseen state of matter": They managed to get LaTe3 to a state it usually isn't in, but there are similar materials (replacing La with another element) where that transition happens at a specific temperature. As usual the headline is disappointingly misleading.
(*):
The crystal structure they work with has one direction that is "special" compared to the other two.