I know he was a masterful tactician, he unified the Greek states and created a huge empire, and won many important battles, but what exactly do people mean when they say he's the most important figure in western history? What about the west changed because of Alexander? Is he simply important since he served as a future inspiration for western conquerors? What were the effects of Alexander's campaigns that actually had lasting and important impacts on western civilization ? Cultural impacts? Economic? Political ?
This is a monstrous question and opinions will vary depending on who you ask, but I'll take a concise stab at it.
If I had to point to one thing that transcends Alexander's military prowess, it would have to be that the fact that he spread Greek culture and language as far as he did. The wars of the Diadochi and the fragmentation of his brief but spectacular empire do not overshadow the fact that the entire area, from Pella to Multan, was awash with Hellenistic culture. It formed a foundation of common communication, trade, connection between cultures, and familiarity that would persist in the years to come.
A shining example of this is Alexandria (the Egyptian one). Alexandria, alone, was a spectacular testament to what Alexander had enacted upon the land. It became a center for trade, learning, architecture, and science for centuries, and is, in my opinion, emblematic of the change he brought through conquest. Many will argue Alexander was a civilizing force, and while romantic and perhaps overly optimistic, it is easy to make this case. Conversely, one must not forget that with conquest comes destruction - the once-smoldering remnants of Persepolis - the Persian cultural capital - attest to this.
Politically, Alexander transformed nearly all that he touched. Alexander's rise is often cited as the end of Greek military vitality, partially thanks to his father's clever diplomacy and military reforms. With the exception of Sparta, Alexander effectively stomped out Greek resistance (see the story of Thebes as a particularly brutal example), and while it would continue to harry him the further East he went, Greece never returned to a position of regional dominance like it had in the century before. Also, it is worth noting that Alexander's trek into the East was seen (or at least marketed heavily by Alexander himself) as a righteous counterattack against Persian dominance in the century and a half prior. Long had Greece fought Persian kings, and Alexander justified his march eastward by explaining that it was high time for Greece to fight a war of retribution and vengeance - particularly when he (and his mother, Olympias) argued that his own father had been slain by "Persian gold" (the actual details are a bit more shadowy and intrigue-filled than legend would have you believe). This, in itself, had significant political ramifications at the time, and in a sense, united Greece toward a common aim. Let's not forget, however, that Macedonians were considered barbarians by the Greeks themselves, and Athens and its shrewd politicians would forever remain a thorn in Alexander's side.
That said, Alexander shaped the region for centuries to come. In the wake of his empire, new kingdoms arose (the Diadochi) that would have vast and fundamental implications for every region they claimed. Egypt (helmed by the Ptolemaic dynasty, from which Cleopatra would descend), Seleucia, Anatolian successors like the Pontic factions, Bactria, and Persia were all glossed with Hellenism, and while these kingdoms and cultures would eventually diverge, they did so for a great while possessing an Alexandrian flair.
It must also be mentioned that as much as Alexander gifted Hellenism to the world, so too did he prepare his kingdom for a spectacularly bloody civil war with multiple sides in which few emerged better than which they had started. One of Alexander's great failings - and this is key to understanding much of what followed - is that he neglected to name a successor as he lay ill upon his deathbed and didn't father a widely-accepted, legitimate Macedonian heir. This would have horrible, destructive political consequences as the Successors vied with one another for military and political supremacy.
His cultural impact is cross-cultural, as well. I don't think it's too lofty to suggest that there has never been anyone like Alexander - that much is clear and evident in just how far his persona, myth, and legend pervades stories and folk tales throughout the Hellenosphere. Heck, even Shakespeare waxed poetic about Alexander. For centuries he was the ideal, and honorifics would take after his own - "the Great". He was, and some would argue, remains, a shining testament to personal achievement (thought Dante might argue about that), and conquerors, statesmen, and leaders for centuries to come would lament that in their late age, they had not accomplished as much as Alexander had in his thirties. For centuries upon centuries, kings, statesmen, and politicians would claim they possessed Alexandrian trinkets, or that their city was the true burial place of Alexander. The weight his association carried was immense.
I hope this suffices as a brief, topical bullet point list for you. Truly, this is a subject that could an entire lifetime of study. The man was a spectacular, triumphant enigma that has spawned dozens of interpretations, most of them positive and some of them negative, and his legend and adventures have certainly echoed through history like few others, if any.
EDIT: Because I've received some counterpoint with regards to my stance on Alexander's "hellenising", I will write a brief response. I did not mean to suggest that Alexander brought Greek culture with the explicit intent of shifting or creating a new Greek world. /u/mythoplokos wrote a good critique against the perceived "hellenising" mission of benevolence, a weak notion I feel is rooted more in the 18th and 19th century theory that the Western powers could justify their imperialism by envisioning themselves as bringing the gift of civilization to indigenous peoples. Alexander's history can be twisted into this idea, and I believe it is the reason it still persists today in some fashion.
I will, however, stand by the assertion that the spread of Greek culture and language would not have taken place in the manner it did without Alexander. If we do away with the term "hellenising", what then takes its place? We can debate the merits of the word and the theory once behind it, but the essential fact is that Alexander came, and with him "hellenic" culture, Greek language, and all that was associated. Yes, it was perpetuated by the Diadochi kings, but does that fundamentally change the discussion? In my opinion, no. Were it not for Alexander, they would never have been there in the first place.
SOURCES: My apologies for not getting this up sooner, as I had just enough time to get this up before dashing out the front door! These books are well-regarded works and I would recommend them as a good starting place for those wishing to know more.
Alexander the Great - Robin Lane Fox (an excellent, in-depth analysis of various sources and opinions. Fox's analysis of the events leading up to Alexander's ascendancy is first rate, and the entire work is worth reading multiple times.)
Alexander the Great - Paul Cartledge (I would recommend reading this second)
The Generalship of Alexander the Great- JFC Fuller (Fuller presents a shrewd, well-sourced look at Alexander's campaigns and political maneuverings. Worth reading for those whose interest is not sated by the two substantial works above)
Who is saying this?
Western civilization, like other civilizations, had many greats. You can't really say one is the single most important figure. Especially when alexander didn't really leave the intellectual traditions of western civilization like the pre-socratics, plato/socrates, etc. Alexander's legacy was more as a military leader and his influence was felt more in none western nations/civilizations he conquered.
Where did you hear this and in what context? I've heard historians say that he, along with likes of genghis khan, were the most consequential military leaders in history. Certainly there is no denying alexander's military achievements which, until genghis khan arrival, was unparalleled.
It's really hard to even quantify history to make an objective judgment of "single most important figure in western civilization". Certainly western civilization didn't begin with alexander and it didn't end with alexander. But his legacies ( such as alexandria where euclid set his Elements ) would help advance western civilization. But I'm now sure how you would even compare alexander to plato/socrates/aristotle.