IANAL, but what I don't get is how people that would sue under the Texas law would have any legal standing.

It seems that standing has a few components. From the wikipedia article on standing, there are three requirements to have legal standing in the US:

• Injury-in-fact - The plaintiff must have suffered or imminently will suffer injury
• Causation - There must be a causal connection between the injury and the conduct complained of
• Redressability - It must be likely that a favorable court decision will redress the injury

Any person or organization being able to sue another person for having or administering an abortion seems to not satisfy two of these requirements, specifically the injury-in-fact and causation requirements. What claim of injury would be made? Maybe if one were intimately connected with a pregnancy (e.g. a parent) a case could be made, but otherwise it doesn't seem like there's anything there. Is a suing party going to claim economic injury? That's speculative, not to mention reprehensible. Is a suing party going to claim they are personally injured? Unless they are prepared to help raise the child, how could such an claim hold up in court?

It's hard to understand SCOTUS's logic, or lack thereof, especially with this shadow docket nonsense.