There is no magic method that makes it work. There are a combination of factors:

1. Patience. Most of the time you're not going to get a reasonable shot. Be prepared to spend a lot of time waiting. While you're doing that, watch a lot and see how your subject behaves. Once you learn what it does when, you have a better chance of approaching closer.

2. Take anything that looks like it could be good. Sometimes there are subtle things that make a shot good or useless that are not apparent when you're thinking about 20 other things and ballancing in some uncomfortable pose, lying in the muck, or being eaten by mosquitos (they actually want to get close to you). There is no point shooting obvious crap, but otherwise the more you have to chose from later, the more likely you find one good shot in the bunch.

3. Get the critters used to you. This simply takes time. See #1.

4. Don't try to get closer all at once. That is pretty much guaranteed to make anything that can run or fly away run or fly away.

5. As you do get closer, never go straight in. Most things get scared off by a large "eye" moving right at them. Head somewhere off to the subject's side, but still get slowly closer in the process.

6. As you get slowly closer, take this is better than anything else I've got so far shots. Sometimes with some good cropping they may be useful if you don't need to display them at large size or close up.

7. With very mobile insects, it's often better to find their routine and wait for them to come to you. Insects don't seem to think of you as a problem if you appeared to be part of the fixed landscape when they got there. And, "there" can be quite localized from your point of view, like 5 meters away from where the insect was previously busy around a clump of flowers.

8. For insects, try to pick cooler times. You don't always have that option, but insects in general move slower when it's cooler since they are cold blooded.

9. Luck. Good technique makes the outcome more likely, just like bad technique less likely. But, in the end there is still a considerable element of chance. Sometimes, even most of the time I'd say, you get skunked. That's just the way it is. Wildlife photography isn't for everyone, especially the impatient. It also helps if you actually like to be outdoors away from crowds.

Here is a case of a rather fast flying insect:

This is Hyles lineata, commonly known as a hummingbird moth or sphynx moth. I saw this moth making the round of a small hillside of scarlet gillia flowers. It didn't mind me following it too much, but didn't stay long enough at a flower for me to set up and focus (all manual back then). Eventually I noticed a pattern it was sortof following. I set up on a flower it was going to get to and waited. Eventually it came and I got a decent shot. No, I'm not going to show you the other 20 shots that didn't work out so well. Although it was mid afternoon, the sky was overcast. I used a flash on a extension cord held by the front of the lens so it would dominate the exposure to get faster effective shutter than the cloudy light and near macro would otherwise allow.

Sometimes, actually most of the time, it doesn't work out that well:

This was one of those better than what I have so far but not really what I want shots. This is a crop from a rather larger frame. You can see the film grain even at only 600 pixels maximum dimension. I still like this shot because it shows a pepsid waspe mimic eating a moth, something the real pepsid wasps don't do. I have been back to this area in Arizona several times trying to get a better shot, but haven't even seen another pepsid wasp mimic since then. I just have to keep telling myself to feel lucky having gotten this.

Be prepared for lots of these:

You get set up, in focus, reasonably close to get good resolution, good lighting, framed, and you take the picture. Then between the time the brain says to take the picture, your finger pushes the button, and the camera actually does it, the little bugger buzzes off. Yes, they can do that in a fraction of a second.

However, keep on going. Sometimes everything comes together and you get something you actually like and are willing to show others without pretending someone else took the picture:

These Diogmites are quite active. Fortunately they aren't bugged too much by photographers as long as you don't get within some magic radius. I found this one minded me less and less as I hung out by the same manzanita bush it was perched in. I don't know, but I'm guessing this was it's inactive period and it didn't want to bother moving too much. After about 30 minutes of it slowly letting me get closer each time before flying off, I could get close enough with a 135mm lens and extension tubes to get a decent picture. If I had been more prepared and willing to spend a few hours there, I could probably have done better. Again, see point #1. As it was, I got caught in one horrendous downpour and thunderstorm before getting back to the car later.