Quoting from the paper you refer to:
These connections form before the retina can respond to light, but at a time when retinal ganglion cells spontaneously generate highly correlated bursts of action potentials. Blockade of this endogenous activity
By "endogenous" here they are referring to the spontaneous activity generated before sight begins mentioned in the previous sentence, which organizes the circuitry for vision. It's endogenous because it's not triggered by an external stimulus like light, it's internally generated.
More generally, people tend to refer to "endogenous" activity that is not specifically related to stimulus-evoked activity that an experimenter can manipulate. Sometimes this is also called "noise" though there is debate about whether this term should be used (not that the people using it actually think it's truly noise).