You see a lot of this on Biggest Loser, hodling a wall sit, a push-up position, etc. What does this do for you and how?

I can understand the mechanism of repetitive action building muscle tissue and the idea behind cardio, but if you stay still in an anaerobic pose hold, what does that help with and how does it do it?

It helps strengthen and tone muscles, specifically stabilizer muscles.

Holding a push up position or plank requires a lot of core strength. All the muscles in the core, back, shoulders and legs are tense. The longer the pose is held for the more fatigued the muscles get from working. Combination of this and lifting can give great results.

Simlarily a wall sit works the leg muscles if done correctly.

Yoga works off of this theory or strengthen while stretching and holding different poses.

right. you're working the muscles in different ways- stabilizer muscles, fast twitch like in jumping/plyo, not sure of other types?
for example for legs
then when you combine the fatigue from doing a wall sit and do squats, lunges, etc. i don't have to do as many squats. it can take forever if i'm just doing squats, with no weight or small amount of weight. so this seems to save time. i do the wall sit type position when i'm doing other activities. brushing teeth, doing dishes, watching tv, or squats when i bend down to pick up something. this is just examples for legs. you could fatigue your abs or upper body in similar ways.

HOW though? Anatomically and mechanically how, not the "you do it and it has this outcome" - how does the process act on the body to produce an outcome? Why does it make you tired? Most people can sit up all day without feeling extreme fatigue and pain and that uses muscles to stop yourself falling down but we are not in agony of fatigue doing that (I am, but that's another story!) so is it the unfamiliar action of holding a wall sit that makes it do something? What does it do to the muscle that renders it stronger? What is the process by which holding a pose makes your heart rate change, makes you sweat, makes you tired? I want to get a bit more into it.

I'll try an example. If I were to be asking about how do you bake bread, one answer is that you follow a recipe and you put it in the oven and when you get it out it's bread. The "why" answer is that strands of gluten are elastic but firm and are able to expand due to the action of the yeast, trapping air bubbles inside, where the baking process then solidifies the gluten and other parts of the gluten mixture by baking out the air, thus the bread has risen due to the incorporation of air bubbles into the mixture.

What would be the equivalent "how and why" answer about how holding a fixed pose actually has any kind of result - why is it that your body has to work harder at holding a pose in a wall sit than holding a pose lying down, both are essentially static positions, what is the process that makes one into total inactivity and the other into an exercise?

Muscle activity is the difference. On the floor, your weight is supported by the floor. In a chair, most of your weight is supported by the seat. On a wall-sit, your weight is mostly supported by your leg muscles.

Holding a wall sit requires your leg muscles to hold up your body weight. You do also use strength to hold yourself up when sitting, but those muscles are used to that activity, and therefore have the strength to do it without fatigue (the spine structure also helps on this, and lots of people slouch against their chairs to keep some of the strain off of their backs)...also the weight being held is minimized because most of the weight support comes from the seat of the chair. When you hold a position on a wall sit, for example, you're stressing your leg muscles because the muscle is what is responsible for holding up your weight. Having your back against the wall takes some of the weight onto the wall, but since gravity pushes you downward, most of the weight goes downward as well (into the leg muscles). Specifically, due to the right angle of your legs on a wall sit, the weight is concentrated onto your quadriceps.

Holding a pose lying down, your weight is supported by the floor, not your muscles. Holding a pose where your weight is supported by one or more muscle groups that do not customarily support that weight puts strain on those muscles. Strained muscles develop microscopic tears. The repair of those tears results in stronger muscle.