People don't like failure. They see it as a bad thing. And it can be. It can be terrible. If you let it.
I think of failure as similar to ingesting poison. On the face of it, it seems like a catastrophically bad idea. And we desperately want to avoid it. Who in their right mind would willingly drink poison?
But, honestly, many of us do choose to drink poison. Alcohol is quite literally a poison. But many of us choose to drink it, and some even pay a great deal of money for the privilege. The important principle is not to drink too much of it.
Let's extend this analogy a bit. Let's suppose you buy a big bottle of vodka (I'm not advocating this, but for the sake of discussion, suppose you do this.) It's late September. You decide that by the end of the year, you're going to have finished the bottle.
You can handle this in at least two ways.
One way is to take a shot or two every evening. A shot or two of this poison won't do much—it may warm you up, make you a little more relaxed and euphoric. It may actually help you in a social situation such as a party or karaoke. You probably won't even have a hangover the next day.
Another way is to wait until New Year's Eve—and drink the entire bottle. That will be a less pleasant experience, and you may not even survive it.
The first way represents small, recoverable errors. Low-stakes failures from which you have plenty of chance to recover.
The second way represents a big, single point of failure. It's high-stakes, and you may have no way to recover from it.
Now apply this to the things you need to do. If you're a student, that may mean drafting papers and projects. If you're an academic, that may mean publishing so you can get tenure. If you're an entrepreneur, it may mean developing your idea so it can find buyers.
In any of these cases, you should build in multiple points at which you can make small errors: low-stakes failures from which you have plenty of chance to recover. Take a couple of small shots every day. You might even look forward to these small failures, and they may make you feel more relaxed, euphoric, and sociable. Each small failure gives you feedback and helps you to better understand a path to success.
The fewer points of failure you have, the larger the dose of poison is, and the harder it will be to recover from it. And if you wait until the very end—you write the paper the night before it's due, you keep putting finishing touches on that journal article, you wait until your innovation is perfect before you share it with the world—then you're looking at a high-stakes situation with a high potential for failure and no margin to recover.
So: Get out there, fail early and often, but in small ways. Understand each failure as a piece of feedback that can help you succeed. Seek criticism and use it to strengthen what you do.
And that is what people mean when they say "fail faster."