The "I'm not a satellite engineer, I don't even play one on TV" explanation is:

Most satellites get their power from solar panels. Depending on their orbit, they might be in earth's shadow for part of their orbit. During that time, they'll need to run off batteries. Batteries have a limited capacity, and their capacity degrades over time. You design the satellite to only have the amount of capacity you think you'll need, and a little extra margin.

Satellites have management computers, with various levels of self-monitoring/recovery. Normally if something goes wrong, the self-monitoring will be able to detect this, and either "reboot" automatically, or go into a recovery mode where the ground-team can send up commands to diagnose and reprogram it.

However something went really wrong with this satellite and that system wasn't working.

The significance of the eclipse is that this would be a longer time the satellite was in shadow, and so the batteries might run down to the point where the satellite switches off entirely. Then, when the satellite is back out of the earth's shadow, the satellite will switch back on when it gets power back.

This would, hopefully, make the onboard computers go into a recovery mode, allowing that diagnostics/reprogramming.

It's sort of an extreme version of "Have you tried turning it off and back on again?".