Yoda's point is that like a lot of mental and physical coordination skills, it matters how you think about it. He isn't saying "don't try", he's saying "don't think of it as 'trying'", which is an entirely different statement.
It's actually a very common observation when learning new skills. Initially you 'try' to do it, thinking about how you move, constantly checking yourself, and mess it up. Your conscious motor control is very flexible in what it can get the body to do, but also relatively slow and clumsy. But after repeating the action hundreds of times, the subconscious part of the brain learns it as an automatic movement. You don't have to think about it, you just do it. It's like walking, or running, or writing, or typing, or catching a ball. If you had to think about how to move your limbs into the right position, you would be far too slow, and would constantly make mistakes. The subconscious control is more limited in what it can do, but what it does is done with a grace and speed the conscious mind cannot match, and is commonly not much involved in. You can walk and think about other things at the same time. It's like dancing - if you think about what you're doing - if you 'try' to do a particular dance - it looks awkward. If you let the automatic control take over and just 'do', it looks much better.
This experience is so common with athletes and others who have learnt amazing skills of mind-body coordination - the feeling of that point where the automatic subconscious control learns the trick and what previously felt awkward and difficult suddenly becomes smooth and easy, and the euphoria it arouses - that it has created numerous mystical and near-mystical beliefs around it. Scientists talk about muscle memory, athletes talk about 'The Zone". Eastern martial arts have a concept called 'chi' - a mysterious force that flows through nature, and that if you let it guide you gives you supernatural grace, skill, and power. (Sound familiar?) Taoism has a saying, that you should "do without doing" that tries to get across this same idea. You do it without having to think about doing it. "Doing" is using the planning parts of the brain to control it consciously, which is clumsy and usually messes it up. "Do without doing" means the subconscious does it for you, without your conscious awareness or direction, and achieves the result with a power and perfection your conscious mind cannot match.
"The Force" in Star Wars is fairly obviously modelled (for those who know about them) on Taoist ideas about the Tao, or Chi. Yoda is very obviously modelled on Taoist stories about ancient masters and sages, and "Do or do not, there is no try" is a close approximation to the Taoist ideal of "Wu Wei": action through non-action. It is advice to stop using your conscious control to "try" to do something, and let your subconscious automatic control just "do" it.