The question should be "Why do we use these aricles in this sentence?" or "Why are these articles used in this sentence?" "Workers have been repairing the road for a month" The road is a particular road, versus any other road. They are talking about a particular road that is being repaired. Therefore the definite article 'the' is used. A month in this case is like saying one month. They have been repairing the road for a/one month, or for two months, for three months, etc. It is not a particular month, just in general any one month, so 'a' is used. Versus: "The workers have been repairing the road for the whole month of January." In this case it is a specific month, so you use 'the'. Also note, "have been repairing" not "are been" I hope this helps : )

I think it's just because English requires articles for singular object nouns unless you're using proper nouns. 'The' to specify one particular noun (the apple / the group), 'a(n)' to specify any one of a type of noun (an apple / a group).

When you use "the" in this sentence you are talking about a specific road. If you said "a road" then you could be talking about any road anywhere. When you use "a month" you are talking about a general amount of time. If you said "the month" then you would have to specify which month like "the month of January." Hope this helps.