What is the meaning of Mr Soil words to worker ant?

The phrase "nothing compared to the twig of '93" is similar to how humans describe and compare notable events.

Example:

Person A: "Wow it's a really messy blizzard out there."

Person B: "I still remember the big storm of '87, this is just flurries compared to that."

He's just making the leaf seem insignificant compared to an obstacle that had blocked them in the past.

This is nothing compared to the twig of '93.

This is just indicating that there was a twig much larger back at some time in the past.

The "Something" of "Year"

...is just an expression referring to a memorable thing happening at some time in the past (or possibly the future) but the past is more usual.