Kaiser No one knows why you feel better afterwards. The popping noise can be explained by how manual practitioners like chiropractors take out the maximal range of motion of the joint(s) and this causes the synovial fluid to displace and nitrogen bubbles burst to make a sound.
Realigning the joints is mumbo jumbo, if your spine could get "misaligned" like that, no one could walk down stairs or play football because those stresses would easily rival what the hands of a practitioner could apply. However, there seems to be some sort of neuromuscular "release" of sorts which is poorly understood. When you have acute back or neck pain, the muscles surrounding the area tense up, and this kind of makes things worse (as long as there is no "real" injury that has to be protected). This limits the range of movement in the joint, and can cause pain in an of itself by being so tense and not letting up (perhaps some local hypoxia).
After a "crack", the muscles tend to relax for a few hours. This lets the patient feel relief, and oftentimes you will be instructed to go for a walk, do light exercises or something of the sort. As such, treating the symptom may give you a window of opportunity to actually use your joints and muscles, and it is perhaps this that actually helps you. That being said, this is all very acute, and "cracking" has not been shown to have any real effects over time when compared to placebo/light exercises/time, and most acute back/neck pain resolves itself within a few days to a couple weeks.
Edit: It's hard to be nuanced in a short reply that is meant to be be easily understood. Not all chiros are quacks, chiros are a weird profession that differ a lot from country to country, and with schools of thought. However, the ones treating auras or re-aligning vertebrae are not in alignment with current medical knowledge. Even so, many chiropractors even rarely do spinal manipulations, but rely more on traditional physical therapy. Low-velocity joint mobilizations also seem to give the same benefits as high velocity ones that commonly "pops", so there's that. I'm sure chiropractors from for instance the UK or Australia could shed more light on current up-to-date practice.
Source: Physical therapist in a Scandinavian country