How do you wildlife photographers get so close to animals they are photographing?

Aside from having a 1500mm lens! How are these wildlife photographers getting so close to the animals they are photographing? What is the trick to this?

I approach slowly as this might be threatening. I avoid direct eye contact as this might be threatening. I try to maintain a low profile as this might be threatening. I keep quiet and walk slowly - yeah - you know why by now. I wear a giant chicken suit which the other photographers find VERY threatening.

Know the habits of the subject. Look for them where they are normally found. Patience is a big virtue - not just for an hour but for the whole day or even a whole season. Set out food to lure them close. Frequent the same place and linger so you become part of the background. I'm sure there are dozens more...

Don't fire a gun, shout, jump up and down waving your arms, throw stuff at them or any other things that might make them run or fly away. Bring a 1500mm lens and a tripod.

Depends on the wildlife. Birds, I hide and sucker them with food. I don't go after larger game, lately a dude was killed in Alaska trying to get a macro of a Grizzly's nose, a bad idea. I saw a mother in Yellowstone many years ago, tell her kids to go stand beside a Buffalo for a picture, the kids weren't as stupid. I have a rule, if the animal can kill me, I usually leave it to National Geographic........

Charles covered a fair amount of it.
First you want to know where they frequently go. This means looking for tracks and trails. Also things like crushed grass in the area, under trees and such. This applies to larger animals like elk, antelope, deer, and so on.
Second look for watering holes. Everything needs water so any lakes or streams are good places to start looking for those tracks, you will find them in clearer parts of a stream, i.e no brush at the access. Lakes will be where there are very few or no cattails, and brush. Now I will say that your larger bird like Herons, Storks, and such love the cattail areas.
Third is knowing where the sun comes up and sets in an area, so you can get it at your back depending on when you setup. Dawn is the best time actually. This is when many animals start their day like humans, and one of the first places they hit is the watering holes. After the sun is well up they are no where near them. Watering is dawn and late dusk.
Fourth is concealment. There are several ways to conceal yourself. One and what most hunters, and photogs do is setup a blind with a good view of your target area. Bass Pro Shops, Cabella's, and hunting supplies carry nice little inexpensive blind tents. This is the easiest, but is not the best. I like wearing a Ghilles Suit, this allws you mobility, if you need to change areas without spooking your subject off. Well provided you move slowly and quietly. In tracking and getting close it is not where you step as how you step. Teach yourself how to step on dried leaves, dead grass without making much noise. Also when moving you want to have your gear in an up position, so when you go to use it your movement is down, in the animal world downward motion is safety. Upward motion is danger, just watch birds to see this. When they feel safe they are on the ground, when they sense danger they are in the trees. Also many animals will spook if the birds do, birds are the early warning system.
Also on moving you need to learn to walk semi crouch and not bobbing as you walk. Then there are best times to move. Wind can be your friend if it is blowing away from the subject. Since the grass or foliage is being blown around you are not as obvious because everything is moving. But you still have to be slow and steady. If the wind stops you stop. Also with the wind blowing against you your sound is muffled, and your scent is going the other way. Be aware though that animals can be more alert if the wind is high, high winds agitate them, like they do me some days. If the wind is blowing towards your subject from you then stay still, unless you have used a good scent masker. This is only some of the techniques I use. I do not want to write a 20 page article with diagrams on the subject.