First, there's no 1 answer to the question. People went on this crusade, just like all the others, for various reasons unrelated to one another. Some likely went to get rich, some wanted adventure, some were sociopaths, some deeply pious, etc.

That said, the main authority on this for a long time was Carl Erdmann's 1935 book, then supplemented by HEJ Cowdrey snippet available here and Jonathan Riley-Smith and again here. What these all did, more or less, was give primacy to the intellectual currents of the reform papacy, particularly after Gregory VII (d. 1085). Papal justifications for religious violence, stemming from their conflict with the empire over the episcopal investiture --> the mobilization of lay warriors in the service of a papal cause towards liberating Jerusalem and, tangentially, aiding the Byzantines against the Seljuks.

There are some problems with all that though.

Mainly (though not exclusively), there's the problem of the sources. Scholars tend to overly (though understandably) rely on a specific "canon" of sources about the 1st Crusade -- the chronicles written just afterwards, Pope Urban II's sermon at Clermont recorded in some of those chronicles, the letters of the crusaders, and (more recently) charters given to monasteries by departing crusaders. Briefly, in turn:

• the chronicles are useful but only to a point. Every single one of them was written after the success of the first crusade, in the midst of the euphoria surrounding the taking of Jerusalem by the Franks. From what we understand of medieval history-writing and medieval practices of memory, authors wrote "backwards" -- they started from an event and explained it, including its causes, as a manifestation of God's will enacted in the world. In other words, they wrote explanations for events as what "ought to have happened" or "what made sense to them." This means, consequently, that we really can't trust their versions of Urban II's sermon. It also explains why the versions of the sermon across different chronicles vary so widely. Each author was writing his version of events.
• letters are perhaps more useful, but medieval letters were different from modern ones. Medieval letters were intentionally public documents, intended for larger audiences. This means that letters both before and during 1095-99 were intended to convey a sense and not necessarily accurate, authentic events and reactions to those events. They were responses directed towards consumption by a particular audience in order to elicit a desired response (such as sending help, offering prayers, etc.).
• charters. They're not what most crusade historians think they are. They're often read as rather transparent, offering window into a pious benefactor's thought process before leaving on crusade. No. 11th-century charters were written almost always by their beneficiaries not by the donor. They therefore represent the donor's wishes only heavily mediated/ shaped by the concerns of the monastery receiving the donation. So, they don't say a lot about why people left but do say a lot about why monks thought people were leaving.

So, on to what I think was going on. I think we need to look deeper than proximate causes. We need to look at the social, cultural, and intellectual fabric of 10th and 11th century Europe and consider where the people who went came from and how they were raised -- what intellectually shaped them. And what shaped them was nostalgia. What shaped them was the memory that their class (the nobility) had once held a Christian empire that spanned the Mediterranean world, from Spain to Jerusalem. It was, however, lost after Charlemagne's death, after the empire splintered in the time of his grandsons. It splintered because the Franks had sinned and lost God's favor. Yet, they were still God's chosen people and could get God's favor back. The Old Testament (for them) provided a model of sin, fall, repentance, and redemption that they could follow. They just needed to act in the right way. Urban II, who was himself a Frank, raised in this intellectual atmosphere spoke directly to many of these aristocrats' concerns and pitched his basic message in just the right way. It's no coincidence that the map of who responded matches up so well to the boundaries of Charlemagne's old empire. It's no coincidence that the united corps of diverse crusading armies called themselves, insistently "Franks." Many of them were marching through the past towards Jerusalem, attempting to claim what had been lost.