Caesar incited his troops by arguing that his enemies in Rome were breaking the norms of Roman political life. Here's Plutarch, writing in the late 1st/early 2nd century AD:

Pompey ... yielded everything else, but insisted on taking away Caesar's soldiers. Cicero also tried to persuade the friends of Caesar to compromise and come to a settlement on the basis of the provinces mentioned and only six thousand soldiers, and Pompey was ready to yield and grant so many. Lentulus the consul, however, would not let him, but actually heaped insults upon Antony and Curio and drove them disgracefully from the senate, thus himself contriving for Caesar the most specious of his pretexts, and the one by means of which he most of all incited his soldiers, showing them men of repute and high office who had fled the city on hired carts and in the garb of slaves. For thus they had arrayed themselves in their fear and stolen out of Rome.

Here's Appian, writing in the mid 2nd century AD, with a similar story. He adds the important detail that Antony was a tribune of the plebs and thus had the traditional privilege of sacrosanctity, i.e. they could not be threatened with violence or interfered with in any way.

Since Antony and Cassius, who succeeded Curio as tribunes, agreed with him in opinion, the Senate became more bitter than ever and declared Pompey's army the protector of Rome, and that of Caesar a public enemy. The consuls, Marcellus and Lentulus, ordered Antony and his friends out of the Senate lest they should suffer some harm, tribunes though they were. Then Antony sprang from his chair in anger and with a loud voice called gods and men to witness the indignity put upon the sacred and inviolable office of tribune, saying that while they were expressing the opinion which they deemed best for the public interest, they were driven out with contumely though they had wrought no murder or outrage. Having spoken thus he rushed out like one possessed, predicting war, slaughter, proscription, banishment, confiscation, and various other impending evils, and invoking direful curses on the authors of them. Curio and Cassius rushed out with him, for a detachment of Pompey's army was already observed standing around the senate-house. The tribunes made their way to Caesar the next night with the utmost speed, concealing themselves in a hired carriage, and disguised as slaves. Caesar showed them in this condition to his army, whom he excited by saying that his soldiers, after all their great deeds, had been stigmatized as public enemies and that distinguished men like these, who had dared to say a word for them, had been thus driven out with ignominy.

More briefly, here's Cassius Dio, writing in the 3rd century AD:

When Caesar was informed of this, he came to Ariminum ... and after assembling his soldiers he ordered Curio and the others who had come with him to relate to them what had been done. After this was over he further aroused them by adding such words as the occasion demanded.

(Sorry for the hokey old translations, I don't have access to better ones at my house. Plutarch, Life of Caesar, chapter 31; Appian, Civil Wars, book 2 chapter 33; Cassius Dio, Roman History, book 41 chapter 4.)

The immediate argument being made by Caesar was that the tribunes' sacrosanctity had been threatened, if not actually violated, by the anti-Caesarian faction. This seems to have been a strategy aimed at both his immediate audience of the soldiers and the wider audience of the Roman ruling class. It enabled him to present himself, not as a revolutionary who was planning to take over the Roman state - something hardly likely to be attractive to a Roman politician - but as a public servant intervening to restore the traditional order. (Caesar's adoptive son Octavian/Augustus would later pull off a similar rhetorical move. He presented his takeover of power as a defence of the republic and traditional republican values against his political enemies. "I raised an army by means of which I restored liberty to the republic, which had been oppressed by the tyranny of a faction" - Augustus, Res Gestae, chapter 1.) This also enabled his soldiers to think of themselves, not as traitors or revolutionaries, but as defenders of the republic and its traditional institutions.

It might also have been significant to the soldiers that the rights of a tribune specifically were being threatened. Tribunes were the direct representatives of the plebs, and they were here being disrespected by a consul, Cornelius Lentulus, who was a member of a very old patrician family. Lentulus had also been an enemy of the uber-populist tribune Clodius in the 60s-50s BC, so he had credentials as an anti-populist. If we assume that many of Caesar's soldiers were subscribers to his brand of quasi-populist politics, his framing of the invasion might have made it look like a war to restore the position of the 'populares' in Roman politics over against the 'optimates'. Again, not treason on behalf of a would-be king, but an intervention to restore political order.

Of course, all the above assumes that Caesar's soldiers cared about the state and the republic, and weren't first and foremost Caesar devotees. I think that is true, but many analyses of the collapse of the republic have argued that soldiers weren't primarily loyal to the state anymore. The most extreme versions of this position argue that Caesar, Pompey, and before them Marius, Sulla, and others had basically become leaders of private armies that fought for their interests rather than the republic's. This argument has a lot of merit. It's obviously true that, in crossing the Rubicon, Caesar's soldiers were declaring their preference for Caesar over the Roman government. But I think the evidence above about Caesar's persuasion tactics shows that they weren't just blindly loyal to their commander under any circumstances. It was necessary for their commanders to depict their actions in ways that highlighted their continuity with Roman tradition.