broken off by its own weight
"Broken" is a participle introducing a phrase modifying "bough."
Part of the problem here is that, in order to be understood easily, such adjectival phrases need to be close to the noun that is intended to be modified. In my youth, I would have been told that this was a "misplaced modifier" and told to rewrite so as to place the modifier closer to the noun modified. Every Monday, we would be given sentences from papers and asked to identify any errors and correct them without changing the apparent meaning. In other words, we were taught how to self-edit and to rewrite. But today the stress is on expressing yourself even if what is expressed is incomprehensible gibberish.
So enough of regretting the sorry state of education. If we try to fix just the most blatant infelicities in the mess that you were subjected to, we might get
I heard a fresh and tender bough, broken off by its own weight, suddenly fall like a fan to the ground even though not a breath of air was stirring
It is till a very clunky sentence, particularly because, as you alluded to, the verb of perception makes little sense. No one hears a bough falling. One can see a bough falling. One can also hear the crack of a bough breaking or the crash of a bough landing. One might even hear the rustling of the leaves of a falling bough. But the implication of a falling fan is silence: something that may be visible but not audible.
Even the word "bough" is ill chosen. A bough is sizable. It does not float like a fan quietly to earth. Nor is a bough fresh and tender. What is meant is "shoot" or "sprig," and I greatly doubt that you can actually hear a new shoot drop from a tree.
I sympathize. It must be hard to learn English from texts written by those whose own English is pitiable.