This is an important and valuable question, thank you for raising it. I'll split my observations into two effects:
- Malthusian effects
- Benefits of scale
Malthusian effects
Other responses have referred to Malthusian effects, by which I mean the concern that with only finite resources, the resources will be spread between more people, and each person will have a worse quality of life.
Benefits of scale
Creating another person doesn't only create another mouth to feed. It also creates another source of ideas and creativity.
For example, each new birth has the potential to become another Norman Borlaug (who is claimed to have saved a billion lives through his research).
Even if 999,999 new people fail to come up with a ground-breaking innovation which makes the world better, if the millionth person does, it could allow everyone to benefit.
Of course, the flipside is that any new person could be the next Hitler/Stalin/insert-your-favourite-bad-guy-here.
Are the Norman Borlaugs winning over the Hitlers?
If you believe the data that seem to suggest that the world has been getting better over recent centuries, the answer seems to be yes.
There's also benefits around the fact that niche interests/needs are better accommodated at scale. If 0.01% of the population has a rare disease, and the population is 10 billion people, that's a million sufferers -- enough scale to incentivise scientific research. And if successful, maybe everyone is cured/treated. For a significantly smaller population that disease may remain untreated for a very long time.
Do the benefits of scale win over the Malthusian effects?
I don't think this is obvious, but I'm inclined to think the benefits of scale win.
If we look at recent examples of challenges that humanity has faced, human ingenuity has managed a few good successes (the aforementioned example of Norman Borlaug and dwarf wheat; the cost effectiveness of solar power has improved dramatically in recent years; smallpox eradication; saving the ozone layer). Don't get me wrong, we still have more to do! But that suggests we want more brains, not less.
Furthermore, decisions we make today should be based on how the benefits of scale will work in the future, not how they were in the past. Will we be better able to use our ingenuity to solve big problems in the future? Some would argue that AI will make us better able to explore creative new solutions (not that everyone will agree on this).
Lastly, and this isn't really answering your question, but rather picking up on a comment of yours. You said that the idea that saving lives makes the world better is a "core assumption of the effective altruism movement". I don't think this is correct. EA is a movement built around using evidence and reason to do good. If the evidence showed that saving lives was bad, the essence of EA would be unchanged. Furthermore, lots of the practice would be unchanged too -- a lot of EA activity is not linked to saving lives.
If you care about maximizing total welfare, then you will think adding extra people to the population is better until the point where the harm they do to others (via using up scarce resources) outweighs the good from creating the extra person. I don't know where that point is, but it's not obvious that it is a point where 'everyone is suffering and starving'.
If you care about total welfare, then it is true that for any concievable state of the world, there is always a hypothetical better state of the world in which everyone's lives are only barely worth living, but the population is so large that the total amount of welfare is still higher. This is the repugnant conclusion.
The repugnant conclusion is a problem for people who claim to care about total welfare. But you can reject the repugnant conclusion without being forced to conclude that saving people makes the world worse. For example, person affecting views hold that an act can only be good or bad if it is good or bad for someone. Stopping someone from being born is not bad for anyone, because the hypothetical child does not exist. On the other hand, saving someone who already exists might still be seen as very good. The two things do not have to be treated as equivalent.
Person-affecting views have different problems (the non-identity problem, and giving non-transitive ranking of outcomes). But they avoid the repugnant conclusion without implying that saving people is bad.
Or perhaps I'm missing the point of your question completely, and it is more practical than theoretical. Are you just getting at the practical concern that saving lives will increase the current population, and that at its current level this is a bad thing? Like Ian Turner said, this then becomes a complicated empirical question about what the actual effect of saving lives on the population is. I don't know if the answer to that is clear.
Also, even if you think adding an extra person by creating a new life does more harm than good on the margin in today's world, and you believe that saving a life increases the population, it does not necessarily follow that saving a life also does more harm than good, because saving lives also has other big effects (e.g. if someone who already exists dies then the family and friends who are left behind suffer greatly).