There is the virus replicating itself in a body, which leads to the body being able to spread the virus to others. And there is the body getting sick. The second often follows the first, but the two are not necessarily connected, which is proven by asymptomatic spreaders. In the vaccine phase 3 tests, it was only determined whether people get sick, and the vaccine was found to prevent 95% of people from getting sick. It was not tested whether the viral load in vaccinated people who don't get sick, but still had virus exposure, was high enough for a real likelihood of transmission to others. This now results in those warnings regarding potential transmission even through vaccinated people: scientifically that's a possibility that hasn't been ruled out.
However, the science is the easy part here. It gets complicated once politics comes into play. Pretty much all scientists I've spoken or listened to say that they assume that the spread will most likely be at least severely limited through the vaccines. That's because they don't have any other explanation than a reduced viral load that could explain how the vaccines actually work in preventing sickness, and a reduced viral load would also mean a reduced spreading capability. However, not knowing something does not mean it can't exist: it is not impossible for there to be some unknown way in which these vaccines may potentially prevent sickness without also limiting viral spread. And scientists usually want to be scientifically correct, so they don't go out and declare something as fact that they just carefully "assume" to be the case.
This "known unknown" of effectiveness in limiting viral spread is now actively being used by politics, especially those people coordinating protective measures in governments, as a convenient escape hatch out of a problematic situation. That situation is: how do you explain to someone who just got vaccinated that he/she should still adhere to all the protective measures like universal mask-wearing and limiting personal contacts? Because even though there are rational reasons for doing so (first, the vaccines need two shots and some weeks of time to actually build up protection, and second, if a significant portion of the populace is exempt from all the protective measures and their burdens, this incentivizes the remaining, non-vaccinated part to also "exempt themselves", because constraints enforced on all citizens are much easier to follow than constraints only enforced on half of all citizens) there will be a strong impetus within each individual to not follow protective measures and rules anymore, once vaccinated and thus "protected" from the virus. And besides that psychological effect there's the overarching problem of the constitutional impossibility of enforcing wide-ranging limitations on constitutional rights onto people that is not justifiable anymore, once an individual in question can provably no longer spread the virus. Because of all of this, politicians (rightfully) fear the situation that we're going to be in in a few months time, when a significant part of the populace, but not enough for herd immunity, is already vaccinated. An assumption of a possibility of transmission even with vaccination is a godsend in that situation.
Possible "reduced viral load" by those who are vaccinated simply doesn't mean that that load is below the threshold which makes somebody not infectious. In practice, it could, for example, mean that if such persons spend two hours in the room with you they can still transmit the virus to you, whereas the non vaccinated person with the virus would transmit to you in 15 minutes. So you can't just claim that a person is 100% safe because the load is just "reduced."
Additionally, for the immune system to respond, the infection has to happen first -- the virus has to spread through the cells of your body. We already know that the people are indeed infectious before their immune system response makes the symptoms. The delay in response has to exist even among the vaccinated people.
So what is sought after is a proof of sterilizing immunity, and there's no such still. I've read that the UK plans to evaluate the evidence for that in the following months by tracking the people who get the vaccine, which sounds good.
There is no threshold that makes people non-infectious. Or there is, and it is zero. Because even a single virus instance can potentially infect someone else. It is extremely unlikely to happen, but not impossible.
Because of this, for practical purposes, you either have to arbitrarily set a threshold at which someone is considered non-infectious, ignoring that it is not impossible for that person to infect someone. Or you have to stop talking in absolutes entirely and just talk about probabilities.
When taking about probabilities, one usually can recognize a reasonable threshold. In practice we do exactly that with most of the medicine: there could be some small chance that somebody can have health problems because his body responds unfavorably to the medicine, but if that chance is small enough it is considered acceptable when the potential benefits overweight the potential loss when the medicine is not used, its use is allowed. If the chance is big enough (i.e. potentially too many people will be affected) such a medicine is not allowed to be used, at least with the affected group of people.
The same is with the possibility that a vaccinated person infects somebody else. There is some point behind which it could be said that some vaccine has "sterilizing immunity" even if some small level of viruses could be present somewhere. For the current vaccine, the question is if the viral load in some point after the infection is decreased at all, and if it is, how much.
At the moment, however, it's simply not known if, in this case, Pfizer vaccine provides sterilizing level of immunity, if, then when, and in which percentage of the vaccinated. At the moment more or less we just know that the vaccinated are less probable to develop symptoms. Efficacy of 95% here means only one of 20 vaccinated develops symptoms when exposed to the virus, so we know that it's also probable that at least 1 in 20, even after being vaccinated, could be able to infect somebody else while being in the "pre-symptomatic" phase (as it is believed by the researchers that one transmits the virus before one's symptoms starts). We also believe that asymptomatic are also able to transmit. We don't know how much the vaccine affects the transmission that could occur when a vaccinated person is exposed to an infected one, and then later comes in close contact with other unvaccinated persons.
However, the science is the easy part here. It gets complicated once politics comes into play. Pretty much all scientists I've spoken or listened to say that they assume that the spread will most likely be at least severely limited through the vaccines. That's because they don't have any other explanation than a reduced viral load that could explain how the vaccines actually work in preventing sickness, and a reduced viral load would also mean a reduced spreading capability. However, not knowing something does not mean it can't exist: it is not impossible for there to be some unknown way in which these vaccines may potentially prevent sickness without also limiting viral spread. And scientists usually want to be scientifically correct, so they don't go out and declare something as fact that they just carefully "assume" to be the case.
This "known unknown" of effectiveness in limiting viral spread is now actively being used by politics, especially those people coordinating protective measures in governments, as a convenient escape hatch out of a problematic situation. That situation is: how do you explain to someone who just got vaccinated that he/she should still adhere to all the protective measures like universal mask-wearing and limiting personal contacts? Because even though there are rational reasons for doing so (first, the vaccines need two shots and some weeks of time to actually build up protection, and second, if a significant portion of the populace is exempt from all the protective measures and their burdens, this incentivizes the remaining, non-vaccinated part to also "exempt themselves", because constraints enforced on all citizens are much easier to follow than constraints only enforced on half of all citizens) there will be a strong impetus within each individual to not follow protective measures and rules anymore, once vaccinated and thus "protected" from the virus. And besides that psychological effect there's the overarching problem of the constitutional impossibility of enforcing wide-ranging limitations on constitutional rights onto people that is not justifiable anymore, once an individual in question can provably no longer spread the virus. Because of all of this, politicians (rightfully) fear the situation that we're going to be in in a few months time, when a significant part of the populace, but not enough for herd immunity, is already vaccinated. An assumption of a possibility of transmission even with vaccination is a godsend in that situation.