Nobody knows for certain. That is the honest answer, and any article that opens with a confident one is overselling.
What is established is narrower and still interesting: when dreaming happens, what the brain is doing while it does, and why almost all of it evaporates within minutes of waking.
When it happens
Sleep cycles through stages roughly every 90 minutes. The vivid, narrative, strange dreams happen mostly in REM sleep, named for the rapid eye movements that accompany it.
During REM the brain is about as active as when awake. Two things are notably different. The prefrontal cortex, which handles logic, planning and the sense that something does not add up, is markedly quieter. And the body is paralysed: the brain blocks motor signals so you do not act out what you are experiencing.
Those two facts together explain most of the texture of dreams. The quiet prefrontal cortex is why you accept without question that your house has a room you have never seen. The paralysis is why running feels impossible, and, when it persists briefly on waking, why sleep paralysis is such an unpleasant experience.
Dreaming also occurs outside REM, in a less vivid and more thought-like form, which is one reason the older theories built entirely around REM have not held up.
The leading theories
Memory consolidation. Sleep clearly does something for memory: learn a task, sleep, perform better. During sleep the hippocampus appears to replay patterns from the day, and dreams may be a by-product of that replay, or part of the sorting. This has the most experimental support and does not explain why the content is so odd.
Emotional processing. The amygdala is highly active during REM while stress chemistry is unusually low. The suggestion is that dreaming lets you revisit charged material in a safe chemical environment, which is why difficult things often feel a degree less sharp after a night's sleep.
Threat rehearsal. Dreams disproportionately feature being chased, falling and losing things. One evolutionary account treats this as a simulator for danger, run safely.
Activation-synthesis. The brainstem fires more or less randomly during REM and the cortex, which cannot help itself, builds a story from the noise. On this account dreams mean nothing in themselves and are the by-product of a mind that narrates whatever it is given.
These are not exclusive. Sleep may do several jobs at once, and dreaming may be central to one and incidental to another.
Why you forget almost all of it
You dream considerably more than you remember, probably a couple of hours a night. The forgetting has a plausible mechanism.
Noradrenaline, a chemical involved in laying down long-term memories, is at its lowest during REM. The brain is essentially running with the recorder off. Dreams you do remember are usually ones you woke during or immediately after, which is why a dream is vivid at the moment of waking and gone by the time you have found your phone.
What is left is not a recording either. Memory reassembles an account each time rather than replaying a stored one, which is why a dream gets stranger and more coherent in equal measure each time you retell it. That reconstructive quality is a normal property of memory and not a defect, and it is the same feature that makes two people remember one argument differently, which is worth keeping in mind before reaching for a word like gaslighting.
Dreams feel like footage. They are closer to a story your brain assembles at the moment of recall, from fragments, under conditions where it was not really taking notes.
The body is doing other work at the same time
Sleep is not idle time. Immune function depends on it: sleep deprivation measurably reduces the response to a vaccine, and infection increases sleep. Growth hormone is released, cerebrospinal fluid clears metabolic waste from the brain, and consolidation continues throughout.
None of that runs on a weekly schedule, incidentally. The body keeps a roughly 24-hour clock and knows nothing about the seven-day week, which is a social invention laid on top of a physiological rhythm.
What interpretation can and cannot tell you
Dream dictionaries assigning fixed meanings to symbols do not survive testing. There is no evidence that water means one thing and teeth another, across people.
What does have support is more modest: dream content correlates with waking concerns. People under stress dream about stress. People learning a skill dream about the skill. So a recurring dream may be a signal about what is occupying you, without the specific imagery decoding to anything.
It is also a good demonstration that expectation shapes experience. Someone who reads that falling means anxiety will interpret their falling dream as anxiety, and will remember it that way. The same mechanism, expectation altering what is perceived and reported, is what makes the placebo effect measurable in trials.
Do animals dream
Almost certainly some do. Mammals and birds show REM. Rats running a maze display hippocampal firing patterns during sleep that match the patterns from the maze, closely enough that researchers can identify which part of the route is being replayed. That is not proof of experience, and it is a strong hint that whatever dreaming is, it is not exclusively human.
The short answer
Dreaming happens mostly in REM, when the brain is highly active and its critical faculty is offline. It is probably tied to memory and emotional processing, though which of those is the function and which the by-product is unresolved. Anyone who tells you the question is settled is telling you about their favourite theory.