The tests in multiple regression are "added last" tests. That means they test whether the model significantly improves after including the extra variable in a regression that contains all other predictors.
In your model with no predictors, adding A improves the model, so the test of A is significant in the model with only A.
In a model with A already in the model, adding B improves the model, so the test of B is significant in the model with A and B. But in a model with B already in the model, adding A doesn't improve the model, so the test of A is not significant in the model with A and B. B is doing all the work that A would do, so adding A doesn't improve the model beyond B.
As @IrishStat mentioned, this can occur when A and B are correlated (positively or negatively) with each other. It's a fairly common occurrence in regression modeling. The conclusion you might draw is that A predicts the outcome when B is not in the model (i.e., unavailable), but after including B, A doesn't do much more to predict the outcome. Unfortunately, without more information about the causal structure of your variables, there is little more interpretation available.