Antinatalism is an ethical position that assigns a negative value to birth and concludes that procreation is always morally wrong.
Antinatalists refrain from procreation to spare their descendants from all the suffering that is an inherent part of existence. Many antinatalists are also concerned with preventing the suffering inflicted by humans on wildlife, nature and other beings. In either case reducing suffering is at the core of antinatalist convictions.
That means antinatalism does also not condone any harm to already existing beings: While new life is never worth starting this does not deny the conclusion that already existing lives can be worth continuing. Antinatalists can lead happy lives personally but refuse to procreate due to the suffering their children could experience themselves or cause others.
None of the unborn wish to be born but plenty of the living wish they'd never lived
Procreation is a gamble with the welfare of the future child. Like with any gamble there is a chance for either positive outcomes in the form of creating happy people or negative outcomes in the form of creating unhappy people to occur.
Unhappy people here refers to anyone who concludes for themselves that they would have preferred to not have been born. It can also refer to anyone who suffers greatly from life, e.g. due to horrible diseases or one of the many other short straws one can draw in the lottery of life.
Couples who plan to have children are usually aware of the risks involved with procreation (since such negative outcomes can be easily observed) but go ahead with rolling the dice because they conclude the odds of creating a happy child to be much higher than the alternative.
Antinatalists argue that positive outcomes in the form of happy lives, even if they are much more likely, do not justify taking the risk of those negative outcomes to occur because...
Thus refraining from procreation is the only moral choice because it does not come with any risk to cause suffering to an unwitting third party.
For the potential child birth does not solve any problem that it does not first create
This argument posits that even the happy people from argument I would in all cases be strictly better off by never having been born. It does that by exploring an asymetry between suffering, which is intrinsically bad, and pleasure, which is only good if it serves to fulfill an existing need.
Suffering is intrinsically bad because we can observe all sentient beings trying to avoid it. People go to great length to prevent harms, even if it means missing out on pleasure. In instances where people accept harms voluntarily it is generally to prevent even greater harms, such as unpleasant doctor visits or exercising to keep fit. In all cases people would be strictly better off if they could avoid harm: Were there a way to be fit without exercise or to not get sick people would take it.
On the other hand we can never observe people longing for pleasure they have no need or desire for. A heterosexual person is not sad about his lack of need for homosexual affection for example (and the other way around). That is because the goodness of pleasure is not intrinsic but only instrumental to fulfilling a need in a deprived state. That means not having a need is at least as good as having that same need fulfilled.
Before birth there is no suffering and no unfulfilled needs exist. Being born introduces an unknown amount of guaranteed suffering into one's life as well as an ever-growing number of needs that can never all be fulfilled, making it a "bad deal" for every sentient being to come into existence. In other words procreation is never to the benefit of the future child.
Why would so many say they are glad they were born then? Because they are not actually comparing their life to nonexistence (which is a hard thing to grasp) but to a worse state of existence instead. People can certainly be happy to be in good health or have a loving partner, but only in comparison to a state where they would be deprived of their need for such and not compared to not having a need for health or love in the first place.
The basis for this argument is the fundamental cruelty of nature: Sentient creatures devour each other alive for food or compete over sparse resources in many other ways. Procreating will force your child to take part in this vicious process both as a victim and as a perpetrator.
Although our technological progress has allowed us to live detached from much of the suffering we inflict on other species there is no denying it still happens. Most people will consume animals or animal products regularly during their lifetime. But even someone raised strictly vegan will have no choice but to hurt others: From killing insects by merely walking around or driving, to all the animals hurt in industrial processes of products we depend on or those displaced by the expansion of our civilization.
Hence this argument is the logical conclusion of the vegan argument: If you oppose consuming animal products to prevent animals from being harmed you should also oppose (or at least refrain from) creating new humans which will have no choice but to harm animals during their lifetime.
Antinatalism is not an organized position and as such has no common goals. For most antinatalists it is first of all a personal philosophy they practice themselves.
That said in a hypothetical scenario where all humans would be convinced by antinatalism it could indeed lead to the extinction of the human species (unless existing humans find a way to become immortal, which antinatalism would not be opposed to). In other words extinction is not a goal but can be a consequence of antinatalism. The good news is that in such a scenario no one would be sad about humanity going extinct, since future generations can't suffer from not coming into existence. They won't even notice.
Why should they? It is not their fault they are alive after all. If they are lucky enough to enjoy their life (as a happy person from argument I) they can continue on like everyone else but refuse to create new people, since it is impossible to guarantee they will also enjoy life.
And even for antinatalists that suffer greatly from life (as unhappy people from argument I) committing suicide would not be "simple" at all: Their survivial instinct will do everything it can to stop them. Society will do everything to stop them by outlawing assistance and preventing access to easy methods. Even if those hurdles are overcome a failed attempt carries the huge risk to leave them even worse off than before, e.g. paralyzed or locked up.
This question is often raised in bad faith by people who fail to understand that creating a new life is different than ending an existing one or who, while being quick to flippantly tell antinatalists to kill themselves, are actually not in favor of an individuals right to end their life at all.
No, because killing already existing beings causes suffering antinatalists are strictly against it. In fact antinatalists want no one to have to die at all and realize that the only way to save someone from certain death is to not create them in the first place.
Since animals are sentient and can experience suffering all of the arguments above apply to them in exactly the same way as they apply to humans. However animals are obviously even less likely than humans to understand antinatalist arguments and thus will never voluntarily refrain from procreating. There is no consensus among antinatalists on whether that means we should interve to prevent animals from breeding, for example through forced sterilization.