Well, you already know the proximate cause. The "real" cause is a complicated web of interrelated factors, such as the growing use of violence in Roman politics, the Republic's loss of control over its most powerful figures, etc.

/u/directaction already explained the basic conflict between the "haves" and the "have-nots" at Rome. It's certainly questionable how much of a populist Caesar actually was, and the number of devoted, active opponents Caesar had among the optimates was actually pretty small. And, in fairness, we shouldn't overstate the level of political engagement among most of Rome's poor. But on the other hand, Caesar was tapping into a history of optimates-populares conflict when he crossed the Rubicon, and he knew it: his public justification was the mistreatment tribunes loyal to him had received at the hands of the Senate, echoing the fates of people like the Gracchi brothers and Saturninus. His actual reason, as you say, was almost certainly simply to avoid prosecution and disgrace. So I don't see Caesar's civil war as a class conflict by any means; Caesar wasn't fighting for the poor, the Pompeians weren't really fighting for the aristocrats and the rich, etc. But it's also easy to see it as the latest iteration of the Republic's history of struggle between political outsiders and the establishment, and Caesar's actions and public stances drew on instances of that struggle, plus the general "aristocrats vs. people" historical narrative, for legitimacy.

A crucial bit of background history, too, is that this had been done before. In 88 BCE, when Caesar was around 12, the consul Sulla was divested by political rivals of a military expedition he really wanted, so he simply took up command of his forces anyway, killed the tribunes his rivals had sent to replace him, and began marching on Rome "to deliver her from tyrants." Notably, by the way, once he took control of the city he instituted some pro-aristocratic reforms, although he didn't really care about making them last: he headed to war as originally planned. His enemies promptly retaliated by taking control of the city in his absence (and sort of thumbed their noses at him, burning down his houses and forcing his wife to flee the city). So after Sulla had won his war, he returned to Italy and spent the next year and a half or so fighting in open war with his enemies across the peninsula. Victory left him by far the most powerful man in Roman history; he made himself dictator and reformed the state as he saw fit (with somewhat more lasting effect this time). After a few years, he laid down his power, became a private citizen, and since everyone was still terrified of him, lived his last years unmolested and died rich and happy. Point being, Sulla had twice proven that if your political fortunes looked bleak, you could simply march on Rome, and not only could you survive, you could prosper. Because Sulla set this precedent, Caesar knew that returning meekly to Rome wasn't his only option, and if you assume he guessed better than Pompey did about what the reaction would be if he invaded Italy, he would've known it was his best course of action.

And Sulla's march on Rome was just the culmination of an escalating trend of violence being used in politics at Rome. For most of its history Rome was a very non-violent place (although there are a few exceptions to the traditional narrative that politicized violence was completely unknown before 133 BCE). But in that year, Tiberius Gracchus, a sitting tribune, was killed by the optimates because of his political views, and the culprits did just fine. If one looks at the episodes of violence between that event and Sulla's marches on Rome, a self-reinforcing process is clear. Use of violence in service of political ends demonstrated to others that violence could bring about quick, favorable results; it demonstrated what one could get away with; and each time violence was used, it became more accepted. Normalization and intensification of politicized violence went hand-in-hand, until less than fifty years after Tiberius Gracchus's death, a sitting consul (Sulla) could actually invade Rome to kill a tribune because of a political setback. Furthermore, violence also begat violence more directly: each act left survivors who were eager to get revenge. While Sulla's dictatorship created a new order that by and large kept the peace for a few years, the Catilinarian conspiracy and the mob violence of the 50s BCE proved that violence was still a viable feature of the Roman political system. So, long story short, not only was there precedent for marching on Rome, this was merely the natural culmination of decades of escalating, and increasingly normalized, political violence.

The Republic had also gotten really bad at controlling especially powerful figures like Caesar (and Pompey, and Sulla, and Marius...). Competition for advancement was at the heart of Roman public life, and this norm helped prevent any one person from wielding too much influence, while the divided government limited the official authority an individual could have. Occasionally individuals did become extremely powerful, but their rivals still could, and did, bring them down eventually, precisely because they recognized the authority of the Roman system (for instance, Scipio Africanus didn't resort to force to get rid of Cato the Elder). Nobody was so powerful as to be beyond constraint by the state, and social norms still made it unthinkable for powerful figures to throw their weight around as brutally as Sulla or Caesar would. But the routinization of violence eroded those norms. Men became willing and able to raise armies from their clients alone (bringing such an army to Sulla was how Pompey entered the political scene). Rome's expansion brought the prospect for ambitious leaders to fight and conquer in the East, which promised extreme wealth; it's often wrongly said that Crassus was the richest Roman, but after Pompey's campaigns in the East, there's no question that he outstripped Crassus by a long shot. Such wealth promised great political influence in Rome, and could also ensure the loyalty of soldiers, particularly since Marius's reforms had led to the recruitment of poorer segments of society, who placed a high value on financial security (and since official pay for soldiers wasn't great, plunder given as a reward by a victorious commander was all the more important). It's also notable that the main great men of the late Republic (Marius, Sulla, Pompey, Caesar) all got a leg up on their ambitions by serving a more established, powerful figure. And so, drawing those factors together, it became feasible for a general to ignore or outright defy the authority of the state, or threaten it into submission.

Finally, here I'd like to briefly note that nothing was inevitable. If Caesar had been less ambitious, or less sure that his enemies would've ensured his conviction if he returned to Rome peacefully, he might not have invaded. And if Pompey hadn't been as eager to gain the approval of the Senate and be an example of a good Roman, or had simply wanted to be dictator like his old commander Sulla, he might have invaded Italy on his return from the East. Or neither might have happened, and the big civil war that ended the Roman Republic might have taken place much later. Or not at all. Laying out the background factors is crucial, but at the end of the day the reasons why Caesar's civil war happened when, and how, it did depend heavily on personality and the facts on the ground at the time.

But you can see, hopefully, the interrelationship at work here in the downward spiral of the Republic. These, I think, are the major trends that explain Caesar's civil war, but they all fed on each other.

EDIT: Wikipedia links are just for clarification regarding some less popularly-known stuff; I drew from academic and primary sources for this, which are too varied to be really worth posting here en masse, but I'm happy to provide citations for any particular points.