While it sounds like it's either an Olympic sport, or a bad day at a veterinary clinic, the elephant hurling fallacy is really just a lazy debate tactic. It involves the quick accumulation of several arguments at once, giving the impression that one's view is well supported, when in reality, elaboration is necessary.
A good way to spot the elephant hurling fallacy is that it reads like a summary, rather than an actual argument itself. In the opening statement of a debate, one might summarise arguments upon which they plan to expound, and at the end of a debate, one might summarise arguments which have already been made, but the elephant hurling fallacy presents the summary as the case itself.
This may come in a number of forms, such as the example given in the header image. Evolutionists are often quite given to committing the elephant hurling fallacy, boasting of the "mountains of evidence" they never actually give. Many of them regard the very word "science" as some kind of touchdown against Christians. Yet, when asked to elaborate on exactly how any given field gives credence to Evolution, their confidence tends to waver.
A similar example is the citation of a number of Scriptural references, usually only referencing the verse without actually quoting it (e.g. "the Bible supports my position in Genesis 1:1"), much less explaining how it supports their position.
When one encounters the elephant hurling fallacy, it can be tempting to attempt to respond to as many of the arguments as possible with whatever time you have. Of course, assuming enough detail has been given to respond to. "Science has proven..." is such a weak and ambiguous statement, there's little to respond to. But ultimately, one needn't take much time to respond to any claim. Rather, the onus should be placed on the one making the arguments to elaborate on just one of their claims. Ideally, you want them to focus on what they consider their strongest argument. That way, even if you don't get to addressing the rest of their claims, their case has been adequately addressed.