I had the same struggle, but JUST figured it out! It's activated by Elk and Druid's runes. It only activates when a rune (OTHER than the base light gain that every card has) generates light for you.
So here's a way to get the benefit:
Have in hand Carpet Construct, and Druid. Play the Druid. The Druid's growth activates. Carpet Construct's Weave activates, upgrading the Carpet Construct. You can then play the Carpet Construct, to benefit from it's upgraded light gain.
Other things:
the Elk's "Herd Support" rune activates The Carpet Construct's Weave. Elk can activate the Weave several times in a turn.
The Smith does NOT activate the it.
Druid's Growth activates it, but its Root rune does not.
Druid's Growth activates it even the first time you play it, when it gives you zero energy.
Overall: Its effect is a little weaker than I'd expect for a "premium" card. Edit: Never mind, I found some good synergy. This game has some fun depths!