Condensed version: conquest, and the competition between aristocrats it caused. The main problem is that the Republic was an aristocratic régime whose equilibrium depended on a relative equality amongst the wealthiest. But the conquests of Rome led to the important enrichment of people who led them (and also to immaterial gains, such as clients and prestige); and the subsequent administration (one could say organised extortion) of the provinces could reap very substantial benefits. These phenomena created a growing inequality.

There the civic factor comes into play: to get an imperium magistracy (which could lead to the substantial gains), you had to be elected by the comitiae. Therefore, “popular” approval (or, at least, being able to buy off the people who oriented the vote) was an important thing, because there was a limited supply of magistracies. Quite naturally, the increasing tension led to an escalation of political competition (blatant corruption, growing indebtedness (to finance corruption), assassinations, crackdowns on rival factions, private armies…). The civil wars of the 1st century BC were the natural expression of this pressure on the traditional framework; and the ascension of a monarch was the only way out of this, by finally establishing the domination of a single man over the aristocracy.

Though I am by no means a scholar on the subject, I do teach extensively on this period of European history and I think I should caution you in the way you phrase your question. You did say "Republic" a period which predates the 'collapse' o the Western Roman world by several centuries. That said, here goes:

Technically, the "Republic" didn't fall so much as it was eaten away from within. As the Republic became more successful, Senators typically became more corrupt, engineering laws and codes to make themselves immune from Roman laws (or some laws for certain periods of time, but that is splitting hairs, IMO). In addition, the Senators began to use their status and power to accrue more land and wealth for themselves, often at the expense of the plebians or Roman commoners. Many of these dispossessed Romans sold their children or themselves into slavery to pay debts and/or lost their farmlands and so migrated to the cities (most notably to Rome itself). The hungry and poor became increasingly restless, violent, and volatile - perhaps another plebian revolt was in the offing?

This problem was recognized as a serious threat to Rome's security, but few Senators were truly motivated to do anything about it, other than to appease the masses as much as possible [short of, you know, giving them land to live on - that'd be crazy, right?].

One notable exception might be the First Triumvirate - Gaius Julius [later Caesar], Crassus, and Pompey the Great - Julius was the 'face' of the Trimvirate and through a mix of conquest, loyal legions, populist rhetoric [such as the promise of land reform], and generous patronage of plebians through festivals and triumphal parades, he became quite popular and powerful. Of course, if you guess that it was largely a show to accrue him (or the Triumvirate) a wider appeal, you may be on to something there.

Nonetheless, when Crassus died, Pompey and Julius had a falling out, with Pompey backing older, conservative Senators who wanted to put Julius on trial for (amongst other things) corruption. Ordered to return sans Legions to face a certain trial, conviction, and unpleasant future, Julius did what only a super-powerful commander of experienced Roman legions would: he crossed the Rubicon with a legion against orders [the famous alea iacta est moment] and essentially deposed the Senate and Pompey, thus installing himself as a "temporary" dictator to see Rome through its crisis. This latter part (the taking of the title "dictator"), IMO, was Julius' mistake - there were a lot of people, particularly aristocrats, who were at least outwardly loyal to the Republic [or at least the veneer of the Republic as it was before the Rubicon was crossed] so Julius Caesar was stabbed to death on the Senate floor on the belief that he was going to make himself king or something.

The conspirators who killed him thought the plebians might rise up and spontaneously defend the conspirators as champions of the Republic, but they were wrong. The Second Triumvirate, famously composed of Marcus Antonius, Marcus Lepidus, and Octavian (later Augustus Caesar) were more cagey about titles, especially after Octavian had eliminated his rivals; Octavian called himself a princeps or "first citizen" instead of dictator. In reality, he ruled fairly directly and the Senate was a shadow of its former power.

The veneer of Republicanism remained until (I think - I'm sure I'll be corrected if I am wrong) Diocletian finally said (I'm paraphrasing here) "you know what, there is no Republic, I'm an emperor." but in reality, the Republic, per se, died long before Julius Caesar's coup d'etat/rebellion.

tl;dr - the Republic never "fell," it just slowly gave in to the corruption inherent in the system and morphed over time into the Empire.

Like GeorgiusFlorentius said, conquest, and the sudden influx of new wealth, did to the Roman Republic what neither Hannibal nor Attila could.

Look at Sierra Leone, with its diamond mines. Look at Iraq, with its oil wells. Or Russia, with its oil wells. If you look at a graph of oil prices, you'll see that when it falls in the 90's, the Soviet Union collapses, but as it rises again we see Vladimir Putin's rise to power.

Money allows dictatorships to afford well-trained, well-equipped, and above all else loyal armies to keep the people down.

Conquest made Rome rich, and it paid dearly for it. A very straightforward example of this would be what Octavian, who would go on to become the Emperor Augustus, did when Julius Caesar died. He seized control of a treasury intended to pay for Julius's upcoming Parthian campaign. He bought the loyalty of all nearby soldiers, then raised a private mercenary army using what was left over, and marched on the city of Rome. Mark Antony had to give all of his soldiers raises to keep from defecting.

This is a gross simplification. There were a lot of other factors, including increasing numbers of Roman citizens and slaves, but money was the biggest. Before Augustus or even Julius Caesar, there was Sulla, who raised a private army and occupied the capital.