Along the years I have been seeing that some members of the EA and Rationality communities are engaged with stuff regarding forecasting - forecasting tournaments, prediction markets and several other terms I am probably not naming correctly. 
As someone who loves both statistics and our psychological biases and fallacies, I am a bit curious - where is a good place to start learning about this? 
No only the hows - but also the whys - is it mostly for fun? To challenge yourself? To benefit from a prediction that may turn out to be accurate? And in such a chaotic world, where forecasting (I assume) only gets harder (and less possible) what's exactly the point?

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https://youtu.be/e6Q7Ez3PkOw

I put together this video series a while ago to help people get started. The book superforecasting is also a good intro.

As another answer mentioned, I have a forecasting newsletter which might be of interest, maybe going through back-issues and following the links that catch your interest could give you some amount of  background information.

For reference works, the Superforecasting book is a good introduction. For the background behind the practice, personally, I would also recommend E.T. Jaynes' Probability Theory, The Logic of Science (find a well-formatted edition, some of the pdfs online are kind of bad), though it's been a hit-or-miss kind of book (some other recommendations can be found in The Best Textbook on every subject thread over on LessWrong.)

As for the why, because knowledge of the world enables control of the world. Leaning into the perhaps-corny badassery, there is a von Neumann quote that goes "All stable processes we shall predict. All unstable processes we shall control". So one can aim for that.

But it's easy to pretend to have models, or to have models that don't really help you navigate the world. And at its best, forecasting enables you to create better models of the world, by discarding the models that don't end up predicting the future and polishing those that do. Other threads that also point to this are "rationality", "good judgment", "good epistemics", " Bayesian statistics".

For a personal example, I have a list of all times I've felt particularly bad, and all the times that I felt all right the next morning. Then I can use Laplace's rule of succession when I'm feeling bad to realize that I'll probably feel ok the next morning.

For a more EA example, see the probability estimates on Shallow evaluations of longtermist organizations., or maybe pathways to impact for forecasting and evaluation for something more abstract.

But it's also very possible to get into forecasting, or into prediction markets with other goals. For instance, one can go in the making money direction, or in the "high-speed trading" or "playing the market" (predicting what the market will predict) directions. Personally, I do see the appeal of making lots of money, but I dispositionally like the part where I get better models of the world more. 

Lastly, I sometimes see people who kind of get into forecasting but don't really make that many predictions, or who are good forecasters aspirationally only. I'd emphasize that even as the community can be quite welcoming to newcomers, deliberate practice is in fact needed to get good at forecasting. For a more wholesome way to put this, see this thread. So good places to start practicing are probably Metaculus (for the community), PredictionBook or a spreadsheet if you want to go solo, or Good Judgment Open if you want the "superforecaster" title.

Just listing here the things I've found helpful and seconding some of them that have already been mentioned in other comments

I'm teaching a class on forecasting this semester! The notes will all be online: http://www.stat157.com/

+1 to all the other resources in these answers, but never underestimate how useful it is to just get started! I keep this link bookmarked, which shows the currently-open Metaculus questions which will close soonest. Making quick predictions on these questions keeps the feedback loop as tight as possible (although it's still not that tight to be honest).

Also, Superforecasting is great but longer than it needs to be, I've heard that there are good summaries out there but don't personally know where they are. 

Also, Superforecasting is great but longer than it needs to be, I've heard that there are good summaries out there but don't personally know where they are. 

I like this summary from AI Impacts.