This is another study that does a good job of showing what’s happening with “waning immunity”.
https://www.biorxiv.org/content/10.1101/2021.08.23.457229v1
Essentially after vaccination you are protected by a high level of circulating antibodies, like most levels of circulation these begin to contract overtime since if they didn’t your blood would just be a goopy mess. The high level prevents actual infection, virus appears and is immediately eliminated before it can get a foothold.
With the change to the delta variant the virus now does a better job of getting a foothold, be it high initial viral loads, and ability to stick, and enter cells. This means you need higher level of circulating antibodies to prevent a foothold, bad when things contract. However with that said you are still prevented from severe disease, hospitalization and death for what will be most likely your lifetime by memory B and T cells.
The memory B and T cells are still rising after 6 months to extremely high levels meaning that if you are exposed to the virus you will pump out antibodies again within days. That’s how you are protected from the bad outcomes but not necessarily cold like symptoms etc… The idea of the third shot is to boost the circulating antibodies to a level that once again is protective from ever getting infected. Anyone that tells you that you will need another booster, etc… has no idea because every vaccine is different when it comes to how long protection will last. Also the spacing of the initial vaccine was probably wrong where it provided higher protection for a shorter period of time instead of less initial protection and more longevity.
This is especially true because the paper shows that the response is not homologous but has a ton of variability between individuals. One person may be protected for a long time, others much less so, we know certain groups this is especially so like the elderly.