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.