after six months, the effects of antibodies produced by COVID vaccines decrease.
As I'm not seeing any of the other answers mention this, the 6-months thing probably comes from the Israeli preliminary data announced in June...
Vaccine effectiveness in preventing both infection and symptomatic disease fell to 64% since June 6, the Health Ministry said. At the same time the vaccine was 93% effective in preventing hospitalizations and serious illness from the coronavirus.
On the other hand, Pfizer has yet to publish more on this (than that)... but they did mention this again this month.
Pfizer did not release the full set of Israeli data on Thursday (July 8), but said it would be published soon.
On the other hand, some epidemiologists have been a bit skeptical about the claim since it comes on the backdrop of Delta strain being more transmissible... and the outbreaks in Israel have been schools which had the least vaccinated populations (as the vaccines were approved/administered later to the youngest...) And the regulatory agencies of other countries/regions haven't been too excited about 3rd booster shots:
The European Medicines Agency on Friday (July 9) said it was too early to determine whether more than the two shots that are currently required for most approved COVID-19 vaccines would be called for, saying it was confident for now that the established regimen was sufficient.
Likewise the [US] CDC was "not sold" on the idea "at this time".
As I'm looking at more news, the 6-month thing could also be in ref to this UK study published [in preprint] at end of June, but mentioned on the BMJ news...
In a preprint, published on 28 June, researchers from the University of Oxford reported that extending the interval between the first and second dose to 45 weeks resulted in higher antibody titres. They also found that a third dose given 44 to 45 weeks after the second increased antibody titres further, and that adverse events were lower after the second or third dose than after the first. [...]
For the booster (third) dose, 75 participants who had their first two doses with an interval of eight to 16 weeks were assessed. The preprint reported that antibody levels 28 days after the third dose were significantly higher than 28 days after the second dose—3746 tIgG EU and 1792 respectively.
The team also found that binding antibody titres to the beta variant “increased significantly” after the third dose, while neutralising antibody titres following the booster were higher than those after the second dose against alpha, beta, and delta variants.
In the UK, the government has said it plans to roll out a booster vaccine at the start of this autumn, in order to protect the most vulnerable ahead of winter.
(Side note, there's a separate Q here about the claim on the increased interval between the first two doses. That isn't as black-and-white as the antibody count suggests, as the cellular response was lower with increased interval.)
Another side note, Turkey appears to be following the UK example on this, which seems to have planned roughly for the same group(s) to receive the 3rd dose, over 50 etc.
At first I didn't see it mentioned in the other answers, but it seems it's the [same] study from Accumulation's answer being referenced here. And its authors were themselves a bit conservative in interpreting the results:
‘It is not known if booster jabs will be needed due to waning immunity or to augment immunity against variants of concern,’ says Associate Professor Teresa Lambe OBE, lead senior author for these studies. ‘Here we show that a third dose of ChAdOx1 nCoV-19 is well tolerated and significantly boosts the antibody response. This is very encouraging news, if we find that a third dose is needed.’