From here:
While Alexander ruled Greece, the Celts dared not to push south near Greece. Therefore, early Celtic expeditions were concentrated against Illyrian tribes.[3]
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In 335 BC, the Celts sent representatives to pay homage to Alexander the Great, while Macedon was engaged in wars against Thracians on its northern border. Some historians suggest that this 'diplomatic' act was actually an evaluation of Macedonian military might.[3] After the death of Alexander the Great, Celtic armies began to bear down on the southern regions, threatening the Greek kingdom of Macedonia and the rest of Greece.
it sounds like it was a multi-generational effort to invade the southeast. What was the reason for such a long term pressure to move out of the homeland (which seems to be the La Tene culture? centered in Switzerland and Austria?) and eventually take on what at the time was a state that even they considered too strong to attack for the time being? Political situation in the homeland? Climate pressure? Population growth, i.e. not enough land for everyone to stay home? Sudden adoption of some military technology (material? tech? social organization) that made them go, "Hey, I think we have what it takes to take on these rich Mediterraneans now that we couldn't before"?
Celtic migration into the Balkans dates from the fifth century BC so by the time delegations were showing up to meet with Alexander during his Thracian campaign various Celtic peoples were numerous in the middle Danube region. We don't know exactly what pushed the Celts into this area but we do have some idea of why they decided to push down into Greece. It seems that the Celts saw the chaos that afflicted the Greek world following Alexander's death as something they could exploit in the service of plunder. Macedon had long served as a strong frontier against Balkan invasion but with Alexander's successors waging constant wars with each other an opportunity to gain access to Greek treasure presented itself. Knowledge of Greek wealth may have got to the Celts through contact with the Greek cities on the Black Sea coast.
But Greece was not defenseless during the successor conflicts. A Celtic expedition against Macedonia was repulsed in 298 by Cassander, its formidable king. We know little of this incident but it seems to have kept the Celts at bay. While Macedon was far from stable for the seventeen years until the next invasion it was fortunate to have a number of strong and energetic kings capable of keeping the realm together in the face of external onslaughts.
This all changed in 281. By this point the king of Macedon was Lysimachus. This was an old but very experienced man who, having started out as the governor of Thrace (since 323), knew how to manage a frontier. He was, by all accounts, a cruel and heartless bastard but he got the job done. But in 281 he went off to war in Asia Minor against a rival dynast named Seleucus and got himself killed. Seleucus moved to take his throne but he was assassinated by a man named Ptolemy Ceraunus, a freebooting adventurer who was a son of Ptolemy of Egypt. His hastily organized coup threw the kingdom into chaos and the Celts took advantage. Multiple Celtic bands swept southward and Ptolemy tried to intercept one of them. They routed his force and decapitated the usurper before apparently deciding to go home.
But one of the Celtic leaders, Brennus, wanted to keep going. He persuaded his fellow chieftains that they should exploit the instability they had created by continuing to push south. According to Pausanias
But Brennus was a powerful influence on their general councils and individually on all the men in authority, calling for a campaign against Greece, and pointing out the weakness of the Greeks at that time, their great public wealth, and the even greater wealth in the sanctuaries in dedications and in coined silver and gold. He persuaded the Gauls to march on Greece
If Pausanias is to be trusted, then we have a clear motive for the Celtic push into Greece. Given the situation in Greece at this time Brennus was making a very good point. With Ptolemy dead there was no clear leader of Macedon as a string of ineffectual claimants took the throne but with little effect. Potentially stronger leaders were busy elsewhere or simply lacked the means to take control. Seleucus' son Antiochus was far away in the east and could do nothing against the threat. The famed Pyrrhus of Epirus was off in Italy fighting the Romans. Gonatas, the son of the old Macedonian king Demetrius, was holed up somewhere with a handful of mercenaries. The Greek city states were in a bad way, following years of war they had seemingly little capacity to organize a viable defense. This was the chance the Celts had been waiting for. Now was the time that they could push south and help themselves to the ill-defended riches of Greece.
The Greeks managed to fend off the worst of the invasion by putting together a coalition of city states against the Celts, most famously putting up a successful defense at Delphi. While the Celts were repelled from Greece they did not simply disappear. While some returned home to the Danube valley others decided to cross into Asia Minor. There they managed to extract from the local kings a kind of surrogate homeland called Galatia. They would be a fixture in the country for some time, often serving as mercenaries.
I hope this has been somewhat informative and interesting to read. For more on the Celtic invasion you can read Pausanias' account of it here. Pausanias was a 2nd century AD Greek geographer whose Description of Greece gives all sorts of interesting insights into the various regions of the country. He was probably going off the history of Hieronymus of Cardia, a writer who lived through much of this turbulent period. For a good secondary source on the Celts see Lords of Battle: The World of the Celtic Warrior by Stephen Allen.