The reason that Augustus (and subsequent emperors) tolerated the Senate was because doing so gave them legitimacy within the context of Rome's unwritten "constitution." It's important to remember that the Roman Republic was never abolished by the emperors, nor was the Roman Empire a new political entity that superseded the former. To understand better Augustus' motives for keeping the Senate and why he gained legitimacy from doing so, let's compare his rise to power with that of Caesar.

When Caesar defeated Pompey in the latest of Rome's late republican era civil wars, it was clear that the old system was breaking down, and that the power of the Senate and the old patrician class was being subverted by powerful generals, who through leading campaigns overseas could amass great wealth and win the loyalty of their soldiers for themselves, not the Senate. One incident you have probably heard of is the famed offering by Mark Antony of a crown to Caesar upon his triumphal return to Rome. Scholars speculate that this stunt may well have been orchestrated by Caesar himself, as it is unlikely Antony would have made such a potentially dangerous offer without the approval of Caesar. It may have been Caesar's attempt to judge, by the attitude of the crowd witnessing the spectacle, whether they would in fact support him taking the title of king and restoring the monarchy in Rome. Ultimately, he apparently felt that, despite his great personal popularity, they would not approve, but the incident was enough to terrify the hardline republican faction in the Senate into moving against Caesar, ultimately assassinating him shortly thereafter.

Augustus, as Caesar's nephew, adopted son, and political heir, learned from the latter's mistakes. Augustus recognized the fact that, while the average Roman citizen might be largely excluded from political power by the patrician elite that dominated the political system, they still felt a great emotional attachment to the idea of the Republic; he also recognized that the Senate was in great fear of being liquidated or dissolved by men like Caesar, and that Augustus needed to placate them if political stability was to be maintained in the long run. What Augustus ultimately did was assume the de-facto position of monarch, but unlike Caesar, he made sure to do so while making every attempt to present his position as within the constitutional framework of the Republic. Following his final defeat of Antony and Cleopatra, Augustus made a great show of appearing before the Senate and laying down his imperium, relinquishing his armies and provincial commands to return to life as a private citizen. This was a stunt of course, but it gave at least the appearance that he recognized the authority of the Senate over him, and was not aiming at subverting their position. The Senate then voted to grant him several new honors, which he "reluctantly" accepted at their insistence. These included a provincial governorship of several key provinces, conveniently the ones where the largest number of Roman legions were garrisoned. This, combined with the great wealth he had inherited from Caesar and the patron-client relationship he had established with numerous senators and wealthy plebeians, ensured that Augustus possessed what was effectively dictatorial power, but with the appearance that this was all only with the approval of the Senate and within the parameters of the Roman constitution. Thus, Augustus' decision to preserve and flatter the Senate with the appearance of respect and deference granted him legitimacy. The Senate had its status and traditional privileges respected by Augustus, and his maintenance in theory of the Republic won him approval from both the Senate and the average citizens of Rome.

For more detailed reading on this, I'd recommend Adrian Goldsworthy's Caesar and Anthony Everitt's Augustus, which are great and easy to read.

The senate was essential for the running of the empire, and to the emperors themselves, for a variety of reasons. First and foremost, the senate symbolically stood for the great store of Roman authority and power. As an institution with its roots in the very bedrock of Roman history, it commanded serious dignity. We cannot understate the value that the Romans placed on tradition and continuity with the past - to abolish something that they thought was founded by Romulus would be unthinkable. Tacitus puts this speech in the mouth of the emperor Otho during the year of four emperors (Histories 1.84):

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"Tell me, do you think that this fairest city consists of houses and buildings and heaps of stone? Those dumb and inanimate things can perish and readily be replaced. The eternity of our power, the peace of the world, my safety and yours, are secured by the welfare of the senate. This senate, which was established under auspices by the Father and Founder of our city and which has continued in unbroken line from the time of the kings even down to the time of the emperors, let us hand over to posterity even as we received it from our fathers. For as senators spring from your number, so emperors spring from senators."

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The last line of that quote is also significant, and I'm going to come back to it later. For now let's stick with the idea of political continuity, because the senate was essential for it. Especially in times of civil war or contests for the throne, the existence of the senate ensured that government of the empire would continue. It maintained its ancient functions of receiving foreign embassies and dealing with the treasury, and the consuls could act as heads of state if the emperor was away from Rome or dead. On that last point, it was the senate that ensured a transition of power from one emperor, and one dynasty, to the next. From Augustus onward, all imperial powers were bestowed by the senate. Of course, some emperors took the position by force and the senate was rarely in a position to refuse even peaceful claimants, but nevertheless its constitutional position as kingmaker was essential. Remember that Rome was never a hereditary monarchy - there was never a law establishing automatic succession from father to (adopted) son. The debates around the accession of Tiberius lay out the problems of this situation at the very beginning of the principate (Tac. Ann. 1.1-15), and it may not have been properly codified until Vespasian won the civil war following Nero's suicide. The so-called Lex de imperio Vespasiani is a contested and fragmentary text, but incredibly interesting in that it shows the senate conferring imperial power onto the new emperor, while also reminding him of the precedents set by his predecessors and setting him firmly within the bounds of established Roman law. Again, it's really important to bear in mind that Roman emperors were not tyrants who ripped power from an unwilling state, nor were they autocrats who ruled alone. They governed constitutionally, within a legally defined political system. This codification ensured that this system continued in a mostly stable fashion for almost three centuries, until the shocks of the third century meant serious changes were needed.

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Pure geography is another reason why the emperors needed the senate. The emperors needed the senate, or at least the educated, experienced class of men that it was made up of, to provide competent governors for their provinces. The Roman empire was a huge and complicated beast, and no one man could possibly govern it alone. Rome's traditional system of governing provinces was to send senators out from Rome for a period of time, usually after they had held the praetorship or consulship. This continued after Augustus for some provinces, and even those that the emperor took for himself were run by legates chosen from the Roman elite. There was an imperial bureaucracy staffed by lower-class citizens - mostly equestrians and freedmen - but this grew quite slowly across the first and second centuries. Even then, there wasn't really any alternative to the senatorial career path (the cursus honorum) as a producer of competent governors with whom the emperors could entrust the provinces.

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Here I want to come back to the last line in the quote above: "For as senators spring from your number, so emperors spring from senators." The emperors kept the senate around because they were, by and large, the same people. At least until the late 2nd and early 3rd centuries, most Roman emperors came from senatorial backgrounds. Even if they didn't originally come from a Roman aristocratic family, once they were in power it was senators that formed the core of the emperors' social groups. They acted as their friends and advisors, and their wives were drawn from their families. Rome remained an oligarchy to some extent, and at least the successful, long-lasting emperors got themselves deeply embedded in the ruling class of the city.

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Hope this answers your question! It's a dense and often dry book, but the best work on the senate after Augustus is still Talbert's 'Senate of Imperial Rome.' Worth a skim through if you want to know more.