I will preface this by saying all views here are my own, and are not necessarily the standard view in historical linguistics, which regards PIE, Proto-Uralic etc. as having emerged ex nihilo as far as we are or can be aware. I disagree with this view, I think that Allen Bomhard's recent work has provided enough shared morphology and lexical cognates with regular sound laws to establish his Nostratic superfamily, minus Korean and Japanese. With that caveat out of the way, let me explain how I think a synthetic/fusional language like PIE emerged from Proto-Nostratic.
First. A clarification on Bomhard's tree: according to Bomhard, first there was Proto-Nostratic, from which Proto-Afro-Asiatic branched off first, then Proto-Zagrosian (aka Proto-Elamo-Dravidian), then Proto-Kartvelian and Proto-Eurasiatic. I think that descended from Proto-Eurasiatic are Proto-Indo-European, Proto-Uralo-Siberian (see Fortescue and Vajda 2022, "Mid-Holocene Language Relations between Asia and North America"-- PUS is the ancestor of Proto-Uralo-Yukaghir and Proto-Inuit-Yupik-Unangan), Proto-Nivkh-Chukotko-Kamchatkan (also a Fortescue-suggested macrofamily), and Proto-Micro-Altaic (ancestor of Proto-Turkic, Proto-Serbi, and Proto-Tungusic), and Proto-Tyrsenian (ancestor of Etruscan, Rhaetic, and Lemnian).
And now, with this perspective in mind, here is my answer to the question. Janus Bahs Jacquet mentioned a typological cycle, and I think that a close reading of Bomhard suggests that, atleast with respect to the particles that became case suffixes and verb endings in Finnish, the Proto-Nostratic ancestors of those case suffixes and verb endings were standalone words, according to Bomhard, and that Proto-Nostratic was a fairly analytic language, and English has actually gone through a full cycle from analytic Proto-Nostraric to agglutinative Proto-Eurasiatic to fusional/synthetic PIE and once English finally loses the -s, -ren, goose/geese plurals and -d past English will become analytic again. To illustrate this development more precisely:
The Proto-Nostratic postpositions *nV genitive, *kV allative, *tV locative, (and more) were just postpositions.
Only the intermediate stage between Proto-Nostratic and Proto-Afro-Asiatic allowed the postpositions to sometimes go before the word they modify, becoming prepositions, which turned into prefixes in PAA. PAA also did have suffixes though, from the older postpositions. This is how we know that in Proto-Nostratic *nV, *kV, *tV etc. had to be free-standing to an extent.
In Proto-Zagrosian (PZ), there is a *-n- genitive (found in the Elamite [X]-inni "made of [X]" as well as in the Proto-Dravidian *-n- oblique augment), as well as a *-k(k)V allative (if I am not mistaken from here are the Brahui -kki allative, Elamite -ka/-ki allative) and a locative in *-t(t)V. All of this comes from synthesizing various things I'd read in papers by David McAlpin.
Proto-Dravidian (PDr) has a very interesting reflex of the PZ *-k(k)V allative, found in a PDr *-nk(k)V dative -- I interpret this as an instance of "case-stacking", where a new dative suffix is formed by gluing the genitive to the allative to produce a new suffix of identical or similar meaning. This is a phenomenon attested in Hyderabad (India) dialects of Hindi-Urdu. While standard Hindi-Urdu has mujh ko (the accusative/dative for the 1st singular pronoun, formed from oblique mujh + postposition ko), in Hyderabad, we say mereku instead (genitive mere + ku, for ku cf. ko). This is because in Hindustani, the genitive functions also as a case with which you can use postpositions, cf. Standard Hindustani mere sāmne "in front of me", mere liye "for (the sake of) me", etc.
I said the word "glue". We know where this is leading. Yes, Proto-Zagrosian and Proto-Dravidian clearly took those old Proto-Nostratic postpositions and turned them into stackable case suffixes, in typical fashion for an agglutinative language. So here is an example of going from analytic (PN) to agglutinative (PZ, PDr).
Proto-Eurasiatic apparently did the same thing, because Proto-Uralo-Siberian is extremely stereotypically agglutinative. Fortescue (2022) provides many examples of stacking case suffixes and demonstrative stems, e.g.
PUS *-n genitive singular
PUS *-(V)t nominative/absolutive plural
PUS *-n(V)t genitive plural = *-n-(V)t
Many Uralo-Siberian languages also stacked the demonstrative stem (PUS *na/nä) onto pronoun stems, e.g. PUS *ki(nä) "who" and the *-n in the PUS personal pronouns 1sg *mV(n), 2sg *tVn and (identical!) subject verbal suffixes 1sg *-mV(n), 2sg *-tV(n).
Proto-Eurasiatic likely did this kind of case-stacking, or started on the way towards it.
This kind of agglutinative behavior stuck around in Finnish and Inuktitut, but it did not stick around in PIE, another Eurasiatic language which took a very different route.
The *nV genitive is survived in the only PIE hysterokinetic noun stems, which take *-r/l for the nom./acc./voc. and *-n- for the oblique cases. The way this happened is that a formerly agglutinative language simplified clusters of stacked case suffixes until they became fusional morphs. For instance, the *-n- genitive became frozen in the oblique forms of nouns like *wód-r̥ nom./acc., *ud-n-é-s gen. and a new suffix *-(e)s with possessive meaning was reanalyzed as the genitive. So here we see agglutinative (PEur) to synthetic (PIE).
And the final step- synthetic back to analytic again, is something we all know about. It is what English is in the process of doing. We gutted all the old PIE oblique cases, now in Modern English we only have "water", inherited from the nom./acc. form *wód-r̥. To express a genitive meaning in Modern English we use a preposition "of", i.e. "of water". We still have a few case suffixes and ablaut grammar left in English- note the plurals water/waters, goose/geese, child/children. Once English loses that, and sounds more like:
"Today I will go to the store and I will buy three apple because I have three sister and they done each ask me for one apple"
English will have completed one full typological cycle, and it will only have taken 15,000 years (Bomhard's date for PN). Arguably some dialects of English already do this, and as those dialects replace Standard American and Standard British English (I am hopeful they will) English will finally be coming one full revolution of the typological cycle.
My direct answer to your question "how could such a morphologically complex language as PIE emerge--and from what?" is effectively the typological cycle hypothesis, which says languages cycle from analytic->agglutinative->synthetic->analytic. This cycle of course only describes changes in certain morphological elements in Nostratic languages, and is not meant to be taken as "the thing all languages must do all the time". I have merely provided an example throughout language families where this happens, and what kinds of changes can cause a language to change typology. There could very well be other possible changes too, as some other commenters have alluded to.