What they are calling a "pad" might also be called a "contact". The electrode is a half-cell that touches the nervous tissue/extracellular fluid; it's connected with a wire to the "pad", which you'd then use some sort of connector to contact them and convey the signal to an amplifier. So the circuit is:
brain -> electrode -> pad -> amplifier -> ground (brain)
When they write
metal lines interconnecting electrodes and I/O pads
the "metal lines" are effectively wires, and by "interconnecting" they don't mean "connecting both electrodes and I/O pads to (something)" they mean "connecting electrodes to I/O pads".
Here's another paper about the specific electrode design that I think is a bit more clear about their construction and the arrangement of the pads vs. electrodes:
Schuhmann Jr, T. G., Yao, J., Hong, G., Fu, T. M., & Lieber, C. M. (2017). Syringe-injectable electronics with a plug-and-play input/output interface. Nano Letters, 17(9), 5836-5842.
In Figure 1a of that paper (see below), "i" is the electrode and "iii" are the pads, ready for a connector.
You might find things described as "pads" elsewhere that are really just the electrodes themselves encased in something soft-and-sticky; for example, the electrodes in an EKG or rudimentary EEG look something like this:
These hold the electrodes themselves against the skin, usually with a conductive gel placed underneath for better contact, but there's really no relationship to the pads in the paper you linked.