Why does Alexander the Great call Greeks free men while Persians slaves?

There is a lot of truth in it

Ancient Macedonia was a borderline Hellenistic kingdom (border towards barbarians to the north mostly) . As such, it was tribalistic warrior culture, with many customs similar to aforementioned barbarians. Being a member of Macedonian army was seen as a duty, but it was also an honor. Essentially, those who would try to avoid service had low social standing (similar to all Greek city-states). Latter, during Phillip and Alexander, there were lot of mercenaries in the army, and even those who were not could except part of lot (spoils of war), prestige and even some lucrative position in conquered lands. So, Alexander's army was mostly made by volunteers . Also, it is worth to note that Macedonian king was not undisputed ruler. He was more of tribal warlord, commander that commanded the army with the consent of the soldiers. It was not uncommon for those soldiers to sometimes refuse his orders. For example, after the Battle of the Hydaspes, Alexander's troops refused to go any further, and he had to comply. From all of this, we could conclude that man in his army we certainly free to a degree, not maybe in today's interpretation of freedom, but nevertheless far from being slaves.

As for Persian Empire, lot of it depends how would you look at it. It was certainly top-down organization, with claimed divine right of the rulers to rule. Empire was made of many diverse people, and it was simply held by force. Persian army was numerous but not h*m*geneous, in the battle Alexander would often fiercely attack and rout one part (one contingent of troops), others would simply flee because they didn't feel the bond with those being attacked. It is hard to estimate from this distance, but it may look like majority of the troops didn't particularly want to be there. They marched and fought because they had to, and in the moment when military discipline crumbled under Macedonian attack they lost the will to fight. Therefore, from Greek perspective, they did act like slaves: slaves work because of fear of punishment. Once the threat of punishment disappears they don't have any loyalty for their work and their position.

In his comment, Semaphore alludes to propaganda and this is indeed how Alexander's speech (see Arrian) should primarily be viewed. He is tapping into the Greek 'traditions' of resisting foreign domination and of the superiority of Greeks over barbarians (non-Greeks).

Slavery is not really the issue here; rather, it is about subjugation. The peoples the Persian King ruled had offered earth and water, they had surrendered their freedom and, in the Greek view, were thus his (using the word loosely) slaves.

Alexander was also seeking to create unity among the Greeks, reminding them of their historic common foe. Although Alexander had effectively subjugated the Greek states, he had not completely eradicated the democratic traditions which had existed in some states for well over a century.

Even going back to the heyday of democracy in 5th century Greece, freedom was relative: the Greeks had slaves, and women had no say in the running of the state. In this sense, freedom meant males over a certain age having the right to vote and to publicly state their views; it meant freedom from the worst of the tyrants of an earlier age.(1) In practice, slaves and women didn't enter into the equation.

Although Alexander's father Philip II of Macedon had conquered the Greek states (Sparta excepted) by 338 BC,

Macedonian rule did not destroy the polis; it merely curtailed personal freedom....The polis continued to exist during the Macedonian hegemony of Greece....It still elected its public officials and held meetings of its political bodies as inscriptional evidence attests;...

Source: Ian Worthington, 'By the Spear: Philip II, Alexander the Great, and the Rise and Fall of the Macedonian Empire' (2014)

Alexander, though, was emphasizing another aspect of freedom which had long been held in Greece - freedom from foreign domination, or the Persian yoke. The commonly held view among Greeks was expressed by Aristotle who

...believed slavery to be a natural institution, and equally that all 'barbarians' (i.e. non-Greeks) were slaves by nature. It was therefore right and fitting for Greeks to rule over barbarians, but not for barbarians to rule over Greeks.

Source: Peter Green, 'Alexander of Macedon: 356-323 B.C.: A Historical Biography' (1991)

To what extent Alexander, Aristotle's pupil, held the same views is still debated by historians today. Earlier, and somewhat ironically, the playwright Aeschylus had expressed the idea of "the free Greeks and the suppliant barbarians" (pdf) through women in his play The Persians (472 BC):

Early in the Persians the queen Atossa, Xerxes’ mother, relates her dream in which two women are forced to pull Xerxes’ chariot. One woman, dressed in Persian clothes, is obedient to the yoke, but the woman wearing Dorian garb struggles so much that the yoke snaps and Xerxes is thrown from his chariot.

Alexander's meaning, as interpreted by the ancient source) Historia Alexandri Magni of Pseudo-Kallisthenes is made clear here:

Youths of the Pellaians and of the Macedonians and of the Hellenic Amphictiony and of the Lakedaimonians [Spartans] and of the Corinthians… and of all the Hellenic peoples, join your fellow-soldiers and entrust yourselves to me, so that we can move against the barbarians and liberate ourselves from the Persian bondage, for as Greeks we should not be slaves to barbarians.

Further, according to Diodorus Siculus,

While he himself with all his army marched into Caria, winning over the cities that lay on his route by kind treatment. He was particularly generous to the Greek cities, granting them independence and exemption from taxation, adding the assurance from that the freedom of the Greeks was the object for which he had taken upon himself the war against the Persians.

This independence did have its limits, though, but the point was that they were no longer under the yoke of 'barbarians' and this served a useful propaganda purpose even though, in practice, little had changed.

(my emphasis)

(1) The Greeks had a different interpretation of tyrants to the modern idea and did not consider all tyrants to be bad.

Other sources:

Tom Holland, 'Persian Fire: The First World Empire, Battle for the West' (2011)

David Kreider, 'Isonomia - The Greek Idea of Freedom' (University of Montana master's thesis, 1973) (pdf)