The answer is delegation. You’re right, there was almost no way to rule directly from Rome, not the way that nations are able to do from their capitals today. So the Romans divided the empire into provinces, which were ruled by individuals who were simultaneously administrators, judges, and military commanders. Their exact title varied, depending on when in Roman history we are looking, whether it was a senatorial or imperial or equestrian province (these divisions came after the Republican period), and how the person came to hold the position. Going in depth into all of them is fun, but doesn’t really help to answer your question. It is also important to remember that the Romans didn’t “rule” these provinces as strictly as we expect of our nation-state governments today. Roman Provincials (people who lived in the provinces) were often relatively autonomous, and to an extent judged themselves with their own institutions and laws, though with some limits- only the Roman administration was allowed to use the death penalty. The primary duty of these governors was to keep the peace in the provinces, stop foreign invaders, collect taxes to be sent back to Rome, and maintain whatever military presence was stationed in their provinces. Here’s a little article from University of Chicago on provincial administration, if you want more details: https://www.journals.uchicago.edu/doi/pdf/10.1086/471267

For a good example of how this local autonomy and Roman authority played out in the lives of the provincials, there is a pretty common story most western people today are familiar with- the story of Jesus. The Romans didn’t care what Jesus was doing, as far as they were concerned Jewish teachers could walk around preaching all they wanted, they had no problem with him. But the ones who did have a problem with Jesus were the local authorities. It says that Jesus was tried by the Sanhedrin (Jewish religious governing authority) and found guilty, which shows the relative autonomy that this local court had, and they could have punished him in almost any way they saw fit- if they wanted to rip his tongue out, or something. But they wanted him killed, which is why they then have to go to the Roman authorities, and try to convince Pilate- the Roman governor of Judea- to crucify him. Based on what we know about Pilate from Josephus and some other contemporary writers, his reluctance to crucify Jesus and desire to spare him as is described in the Bible may be a bit unlikely- he had a bit of a reputation for brutality, which would result in him being recalled to Rome when they heard of it- but he was willing to order the crucifixion because if the execution of one upstart kept the locals happy, why not? It keeps the local authorities loyal, satisfies the people, and generally just makes Pilates life a little easier.

For your final question, why should they listen to guys they may have never seen, don’t know, and who certainly don’t care about them, I’ve alluded to the answer already- because the Romans had the legions. You are free to believe all you want that you shouldn’t have to bow to Rome, but when there are a few thousand armed men nearby who are specifically tasked with making sure you pay your taxes, you pay your taxes.

It is worth noting, though, that this system was not always perfectly efficient. You brought up that without modern communication, it’s very difficult to rule from Rome across the Mediterranean and much of Europe. This became especially apparent during the Third Century- there was a lot of upheaval, mainly from Generals declaring themselves Emperor and fighting each other in civil wars, but also due to Governors declaring themselves Emperor, sometimes of the whole Empire but other times just over their provinces, and in both cases it was a de facto declaration of independence. The crisis would not end until Diocletian, who realized that, as you said, the Empire was simply too large to be governed by one man in one city, which resulted in him partitioning the Empire, and appointing 3 other Emperors (4 in total), to take care of crises that would erupt in their designated areas. Here’s a good article on the 3rd Century- https://www.worldhistory.org/Crisis_of_the_Third_Century/