For what reason did Dharmakirti argue that absences are conceptual constructions?

According to Tom Tillemans in his article on Dharmakīrti, this particular argument is not of Dharmakīrti’s own invention, but was rather likeley to be inherited from previous writers like Vasubandhu.

Accordingly, “Dharmakīrti has two arguments for the momentariness (kṣaṇikatva) of all that exists”, the first one being commonly known as the vināśitvānumāna:

A key underlying principle of the vināśitvānumāna is that negative facts, such as absences, are not part of the ultimate furniture of the world, but are just fictional conceptual constructions, as they are devoid of causal powers. Equally, a fiction lacking causal powers is not the effect of something else.

It is followed by some of his actual examples:

While it is obviously impossible to deny that hammers smash pots, the absence (abhāva) of the pot, i.e., the non-existent pot, is not an effect, just as other non-existent things (abhāva), like horns of rabbits, are not effects of anything either. Hammers and the like are thus not actually causes of the pot's absence but of it turning into potsherds. That idea is perhaps defensible, in that arguably the mere absence of something—a purely negative fact—might be less real and less efficacious than the presence of other things.

The other argument however, “seems to be largely Dharmakīrti's own invention”:

There is, fortunately, a much better argument for intrinsic momentariness than the problematic vināśitvānumāna. This argument is known as “the inference [of things' momentariness] from the [mere] fact of [them] existing” (sattvānumāna), and seems to be largely Dharmakīrti's own invention, first developed in the second chapter of his Pramāṇaviniścaya. If anything exists and is a specific thing rather than another, it is because of its causal efficacy (arthakriyā), or powers to produce such and such effects.

Both arguments are included in the second chapter of Dharmakīrti’s Pramāṇaviniścaya. So I suggest you look into that book. The critical edition of chapter 1 and 2 has been published, but although Amazon says its in English, I think it regards the Sanskrit/Tibetan edition.

Say you walk into a room and look around and think to yourself, "Ah, this room has no elephant in it!"

The absence of an elephant in that very room is an object of knowledge. It can be known by a valid mind that there is - in fact - no elephant in that room. Since the absence is an object of knowledge it can be said to exist. So how does it exist? In order to answer this, back up and look at how it was known.

The absence of an elephant in that room was known merely by conception and inference on the valid basis of failing to perceive any elephant in that room and depending upon the thought which knew it. It is not a real, essential or inherent absence. How can we know this with certainty? Because if the absence of an elephant were real, essential or inherent, then everyone walking into that room would know that absence and think to themselves, "Ah, this room has no elephant in it!" which plainly does not happen. Moreover, it would be impossible for someone to - at any of the three times - walk an elephant into that room. If someone were to walk an elephant into that room, then this would annihilate the absence. But inherent things cannot be annihilated otherwise they would depend upon something else. Moreover, the absence of that elephant - were it inherent - would not depend upon anything at all.

No, the absence of an elephant in that room exists conceptually, dependently, relatively, in an illusory-like way without even a shred of inherent existence at all. In the same way, were someone to walk an elephant into that room, then that elephant would exist conceptually, dependently, relatively, in an illusory-like way without even a shred of inherent existence at all as would the "absence of the absence of an elephant."

This refutes real Abhava along the lines of what Dharmakirti might employ and is in agreement with Madhyamaka's like Nagarjuna. However, Dharmakirti was not a self-identified Madhyamaka per se and had some apparent disagreement with other points of doctrine, but on this point I think they agree.

If you are talking about Apoha, well that is related but different. That is getting into the epistemology of how nominal things are known from real and inherent - but unconceivable - entities. The upshot is that Madhayamaka's like Nagarjuna had apparent disagreement with Dharmakirti on this point in so far as Madhyamaka's like Nagarjuna did not think that anything at all was real or inherent even if inconceivable.

Hope this helps!