One meaning of "imply" is that one idea to logically follow from another. Like you might say, "The fact that 20 is an even number IMPLIES that it can be divided by 2."
An "implication" is the idea that follows. In my example above, "20 can be divided by 2" is an IMPLICATION of the fact that 20 is an even number.
This may not be the best example, because you could say that the definition of "even" is that a number can be divided by 2, while the idea of "implication" is that it is not just a restatement of the same idea in different words, but an idea that follows naturally, but is a different idea. (I was trying to come up with a very simple example.)
If you've ever said, Well if X is true, then it logically follows that Y must be true because ..., then you're talking about an implication.
So in this case, the writer is saying that if Bahá’u’lláh’s teachings were really followed, that logically and inevitably certain things would happen. I don't quite follow what he is saying will follow, some sort of positive effect on conditions in the villages anyway. Like you might say, "If you really believe that that person over there is your (spiritual) brother, that IMPLIES that you will deal with him fairly and honestly."