Obligatory "I am not a professional historian" (in fact, I'm a biology professor). But my undergrad degree was in classics, so I have a pretty good background in Roman history. Of course I welcome any and all corrections on this from the pros. Here is my attempt at an answer:

You are asking about what is referred to as the "Migration Period". As with most historical events that involve the movement and displacement of vast numbers of people, there is no easy catch-all answer... however, it might be appropriate here to insert a paraphrase of John Dillinger's famous (and maybe apocryphal) line about why he robbed banks: it is where the money is. From the outside looking in, the Roman Empire (RE from here on) was a land of great prosperity and stability. People wanted to go there for the same reason migrants seek out more stable and economically prosperous places to live today. In some cases the movement of people into the RE were violent invasions of the type you are likely picturing (the Huns, for example, conducted a military invasion). In other cases the movements were peaceful and in some cases were even invited resettlements of people with an eye toward obtaining a buffer zone against other, more dangerous peoples, or gaining recruits for the Roman military.

In answering this it is important to clarify a couple of points. First, the principle movements of various peoples into Roman territory occurred over a very long span of time (300s-800s A.D., and really to some degree before this...). These movements were not contiguous, and different groups had different motivations for "invading". I put the term in quotes, because not all movements of outside peoples into the RE were invasions in the military or raiding sense. In some cases tribes entered Roman territory seeking protection or to escape pressure from other groups of people that were, in turn, pressing against their territory from another direction. In some cases the Romans invited outside peoples into their territory to serve as a buffer against more aggressive tribes who were farther away, or to serve as military recruits to defend the borders of the empire. Second, the RE had a series of natural borders defended in various ways (naval vessels on the rivers in some cases, strings of forts, actual border walls in some locations).... but the borders of the RE were massive, and difficult to effectively "hold" due to their immense size and the relatively remote nature of border regions. For centuries, then, the RE used a combination of military power and economic benefits, bribes, hostage taking/holding... in effect crafty diplomacy of various forms punctuated by occasional war, to keep outside peoples outside the border. Eventually, however, internal and external factors combined to break down this delicate balance, and people began coming in. Roman society and military power became more unstable/began to wane, and civil wars weakened the ability of the Roman military to defend the border. Outside groups took advantage of this weakness and moved in.

The Germanic tribes that would become the origin of the "first wave" of peoples moving into Roman territory had origins in Scandinavia and Northern Europe, and by 100 B.C. had moved into the region bordering the Roman province of Gaul. Additional waves of peoples who would eventually become known as Franks, Saxons, Alemanni, and others moved into the Carpathian Mountain region around the same time. For several centuries there was a kind of balance, where cross-border trade, occasional invasions by both sides, etc. occurred without any major incursions into RE territory.

The main period of invasions is traditionally divided into two phases. The first phase occurred ~300-500 A.D., and resulted in the effective collapse of the Western RE and the consolidation of power into several "barbarian" kingdoms (for instance, the Visigothic (nope, Ostrogothic, my mistake) Kingdom of Italy). A second phase occurred ~500-750 A.D. or so, and involved Slavic tribes moving into central Europe, and down into southeastern Europe into territory still controlled by the (Eastern, or Byzantine) RE. By the late 700's-early 800's the former "barbarian" (now well-established and settled) kingdoms built on the ruins of the western RE began to feel outside pressure of their own, in the form of raids and invasions by Scandinavian peoples (the "Viking" invasions).

So, TL/DR would be that a large number of factors motivated "barbarian" peoples to move into Roman territory, including plunder, better economic opportunity and stability, and outright invitations from the Romans themselves for various reasons... but largely all roads lead back to the economic motivation. Rome was where the money was, and people outside knew that. But it was a complex situation, with many motivations at play. I can recommend a couple of good books on this subject. The Cambridge Medieval and Ancient Histories series are excellent.

See: Halsall, Guy (2006), "The Barbarian invasions", in Fouracre, Paul, The New Cambridge Medieval History, Vol. 1: c. 500 – c. 700, Cambridge University Press, ISBN 0-521-36291-1

Halsall, Guy (2008), Barbarian migrations and the Roman West, 376–568, Cambridge University Press, ISBN 0-521-43491-2

As well as Michael Grant's "From Rome to Byzantium: The Fifth Century (1998)"

John Julius Norwich's "History of Byzantium" series, in particular the first volume "The Early Centuries", goes into some depth on what was happening in the eastern portion of the RE, as well as subsequent events after the west had largely collapsed (the eastern Emperor Justinian pursued a policy of reconquest in the 6th century A.D. that is fascinating to read about, but was ultimately unsuccessful and probably caused more harm than good given the amount of money spent on this attempt at restoring lost lands to the west... his troops managed to retake parts of Italy, North Africa, and Spain for a period of time, at great financial cost).

There are loads of books on this subject available, but the ones above are good sources for a start.

Edit: I corrected a few grammatical errors, and added the reference to Lord Norwich's books.