Why did the Roman empire invade the countries?

Wars with the Samnites not very far to the south of Rome led the Romans to seek hegemony over Central Italy and part of the south. Victory over an attempt to invade Italy by a Greek king led to conquest of the rest of the south. Concerns about Gallic threats led to expansion in the north. War with Carthage (in Tunisia) led to control to the western Mediterranean. War with the king of Macedon in Greece was waged out of worries about his alliance with Carthage. This then led to Rome being drawn into Greece as an arbitrators for dispute between Greek states and more wars. Rome then decided to take over. Troubles with kings in Turkey led to expansion in that area and this was followed by conquest of Syria, Lebanon and Palestine. The conquest of Gaul was linked to worries about attacks on Roman territories in the south cast of France. Egypt was conquered as a result of Cleopatra's involvement with with Antony a Roman ruler who got caught in a civil war which he lost and led her to loose her kingdom to the winner of the war. At one point it became as natural for Roman rulers to seek war and conquest.

There are not simple reason for Rome's conquests. it was a process that unfolded over time and the driving forces changed with changing circumstances. In short, there are no simple explanations.