In order to obtain maximum performance on our lifts, we want to achieve the highest level of stability possible. Together with a pair of weightlifting shoes, the belt is the most important piece of equipment we need in order to keep making progress.
The mechanism by which the lifting belt enhances our performance is a more effective muscular contraction around our waist. In particular, the tightness of the belt "draws the tummy" in - meaning a decrease in the space inside the abdominal cavity which results in greater pressure inside it. Greater pessure means a more "solid system" around the trunk and thus a better force transfer between the upper and lower body in our lifts.
The muscles producing a harder contraction as a result of wearing a belt are the rectus abdominis, obliques, transversalis, pelvic floor muscles, as well as the erector spinae. These muscles form a corset around our waist with the predominant job of protecting the lower back from unwanted impact. The belt provides a mechanical resistance against these muscles which forces them to contract harder. A more effective contraction locks the spine into place and prevents a strength leak between the upper and lower extremities. This dramatically increases our ability to lift heavy weights.
In the deadlift, a rigid back is required to prevent lumbar flexion and thus potential injury. A rounded lower back further changes the mechanics of the lift in which the hips start to drop and knees start to bend - shifting the load from the posterior chain (glutes, hamstrings, erectors) to the anterior chain (especially knees). This makes the lift pretty inefficient and thus weaker, if not to say useless.
In the squat, keeping a rigid spine is crucial for maintaining a correct back angle. If the spine is not rigid enough, horizontal forward lean increases, and thus the bar becomes heavier by moving in front of (instead of staying right above) our centre of mass. A back so stable it feels like concrete is also crucial for leading our way out of the bottom position. Together with the movement of hip extension, the spine (or better said: the chest) has to lead our body out of the "hole" to successfully stand up in the movement. The belt creates additional pressure around our trunk to successfully perform this upward phase.
The same principles apply to the overhead press. As our upper body leans back in the initial phase, the belt increases the pressure around our abdominal and back muscles - creating a more efficient force transfer from the ground up via our legs into the arms and shoulders and thus pressing muscles. Greater stability around the trunk means greater numbers on the bar.
In terms of choosing the right belt, I suggest you opt for a 3-inch belt (that's 10-13mm width), single or double ply. A single ply is more convenient to get into place, whereas a double ply distributes the pressure across two lines of holes instead of one line, making the belt potentially tighter.
The pressure should be "strong enough" for the belt to sit tightly around your waist. If it is too tight, it will feel inconvenient and will thus affect performance. If it is too loose, it becomes an item of decoration rather than a tool of performance.
In doubt, tighten the belt to one hole further than what you think you need to tighten it to. I have found that when I draw my belly in a bit more and pull the belt sligthly tighter with a bit more effort, it gives me that compressive feeling I expect from the belt in order to give me maximum stability.
Finally - needless to say - many lifters have overcome plateaus, avoided pain, and improved their technique whenever they switched from beltless lifting to using it in every exercise.
Everyone should wear a belt.
So should you.