The topic you want to research is 'paradoxes of material implication.' That is, you are right to think there's something odd about this formula. But it does not mean that every fact deductively follows from every other. Consider your target string:
Now note that the convention is to define the connectives such that the conditional 'P-->Q' is logically equivalent to '~PvQ'. Hence, replacing each instance of the conditional with the disjunctive equivalent, your target string is logically equivalent to:
~B v (~AvB).
i.e., Either not-B, or else not-A or B. On this formulation, it is clear that this formula does not mean what you suggest it means. Furthermore, this disjunction is a trivial logical truth . There are two ways the world could be with respect to B: either B is true or it is not true. If B is true, this formula is satisfied by the second disjunct. If B is not true, this formula is satisfied by the first disjunct. Hence, it is logically true.
This formula is one among many reasons why philosophers have thought that the material conditional of first-order logic does not capture any interesting epistemically-rich notion of implication. When formulated as a conditional, this leads to valid implications that are counterintuitive, pragmatically odd, or prehaps even false. That is, we usually expect implications to carry some kind of epistemic or pragmatic force, to deliver some information beyond the initial terms. So carrying out this kind of inference, though valid, holds no water in ordinary discourse. Depending on your views about the relationship between semantics and pragmatics, then, there is actually an argument to be made that such "implications" are a kind of nonsense.
Monotonicity is another feature of logic that you may want to look into. Entailment in first-order logic is a monotonic operation, which in this setting means, intuitively, that once something B is proven, the addition of new assumptions will never make B false. One way to think about your formula is through the monotonicity of entailment. Suppose we have some string of background assumptions T, such that T entails B:
T |- B
The monotonicity of entailment ensures that the addition of new assumptions will never prohibit concluding B. Hence, the extension of T by arbitrary new assumption A will not alter the result:
T, A |- B
Many systems are intuitively non-monotonic, such as when we reason in a default way or when we assume that implications should have some relevance to one another. If you're in the United States, then you do default reasoning whenever you pay your taxes. For example, by default you owe the standard amount of taxes for your income level. But the addition of other assumptions --- such as itemized deductions, being the owner of a business, the head of a household, of retirement age, capital gains, student loan interest payments, etc. --- can make those default truths false. We also usually demand that the terms of an inference be relevant to one another. This is, at least arguably, one point of departure between traditional Aristotelian logic and modern mathematical logic. Aristotle's logic exhibits features of a relevance logic, i.e., the assumption that the premises in an argument must have some relevance to one another by virtue of which the terms of the premises can be combined or separated. This is one of the many ways in whcih Aristotle's understanding of syllogisms and deduction differs from our modern notions of validity. How to pin down 'relevance' is an entirely other can of worms. In a Wittgensteinian vein, we might also complain that someone who asserted a conditional like yours in conversation would be failing to make a move in the language game.
First-order logic validates all kinds of at best pragmatically useless and---again, depending on your background philosophical views---perhaps false entailments.
Priest, Graham (2001). Introduction to Non-Classical Logic. Cambridge University Press, pp.12-13 (on material implication)