What is happening here

There are several meanings-
1- It eats the young shoots of plants, and eats leaves from trees
2- It eats something, then shoots a gun and leaves the area
3- It eats the young shoots, then leaves the area

Actually is a bungled punctuation joke--i'll explain that in a moment. This is how it should look if it were not a joke:

The panda eats shoots and leaves.

Panda is the subject, eat is the verb, and shoots and leaves are the objects of the action.

Now, the joke is created by adding a commas (it was badly done in your example):

The panda eats, shoots and leaves.

Shoots and leaves are not only nouns, they are also verbs. So, if you add the comma as i've done above, the punctuation joke is created. The Panda eats (subject, verb), the panda shoots (shoots as with a gun--subject, verb) the panda leaves (meaning the panda departs--subject, verb).



In the example you provided, the joke is ruined because the comma is in the wrong place.

Actually, the comma is pretty insignificant with respect to the central issue of different views and debates about the nature of "grammar". The sentence is typical of an example of an ambiguous one which demonstrates the difference between deep structure and surface structure in Chomsky's "Transformational Grammar". Other celebrated Chomskyan examples (which generally imply speech rather than text), are...

Flying planes can be dangerous.
Visiting relatives can be a bore

Ambiguity is accounted for by two or more deep structures mapping to the same surface structure.

Alternatives to traditional grammar, and transformational grammar, include "discourse analysis" which transcends "the sentence" as a unit of analysis, thereby resolving ambiguities.

Setanta's point about a joke is valid insofar that most, if not all, linguistic jokes might be said to involve sudden awareness of a shift in "deep structure".

It seems to me a joke IS in play here...and is a variation of the verbal one that goes:

"He cooks carrots and peas in the same pot"...which when heard rather than read, sounds like, "He cooks carrots and pees in the same pot."

If the sentence were heard rather than read, depending on how inflected, it could be heard as:

The panda eats shoots and leaves.

The panda eats shoots...and leaves.

The panda eats...shoots...and leaves.

It probably is a joke...but not an especially good one. It belongs in the very, very bad jokes thread.