"In the late afternoon" why does it have "the " here?

Good question, I had to think for a bit and my immediate reaction, I don't think was correct.

However, my understanding would be that the author is talking in a specific manner about a generic afternoon. This occurs a lot in English with time:

In the morning we would wake up and have breakfast

In the afternoon I would play soccer

In the evening I would often go to town

I'm not an etymologist, and am not certain about the origin of this practice, but in each situation, it describes a single, specific 'average day'. It is also tied with using the word 'in'.

As you suggested, you can use this form, or you can also use the alternative:

On late afternoons...

Or even

On a late afternoon...

But these two require usage of the word 'on' rather than 'in'. I'm unable unfortunately to, at this point, explain the why each of these words is used.

I would see such when-indications as

in the morning, in the afternoon, in the evening

as idiomatic adverbials and it is no use trying to analyze the use of "the" with grammar rules. These when-indications have a good rhythm for speaking and the use in this form became the normal thing.