There's a number of reasons, but they all boil down to the same thing: they had good reasons to believe that pitched battle was the easiest and most effective path to victory against the invader.
1. Pitched battle was their thing
The Achaemenid Persian empire was the largest territorial power in the world. In fact it was the largest empire that had ever existed anywhere on Earth. With manpower and financial reserves at that scale, it could field armies and fleets that no one else could possibly match. This is what they repeatedly managed to do against Alexander: not only to field huge numbers, but also to deploy the finest troops from all regions of the empire and use their cash reserves to hire the best soldiers money could buy.
Since the Persians could always assume that they could field larger and better armies than their enemies, their preferred strategy was to provoke their opponents into a single battle in a level plain where the conditions were as fair as possible. In such an environment, the odds were always, naturally, by definition, in the Persians' favour. It helped that open plains were most conducive to the use of the most important shock force of all Persian armies, its elite cavalry. With room to manoeuvre, these horsemen would encircle and trample any enemy that dared to take the field against them.
In every war the Persians fought against Greeks, the Persian approach was always the same: to aim for a decisive pitched battle in open ground. Greek victories tended to happen only when the Greeks denied them the opportunity, either by refusing to leave a strong position or by mounting surprise attacks when the Persians weren't ready. In battles like Kounaxa (401 BC), we see the Persian army in its element, easily surrounding and routing its inferior enemies while using mobility and range to render any enemy heavy infantry useless. At Gaugamela, the Persians had even made an effort to smooth out the terrain so that it would be as clear and featureless as possible, giving maximal advantage to their numbers and cavalry. It was a reliable formula for an empire that knew no rivals.
2. Alexander had no advantages
Imagine you're Darius III for a moment. You rule the largest empire in the world and some upstart hat-wearing Yauna thinks he can challenge your power. Why wouldn't you go out to fight him? Why would you give that upstart the satisfaction of believing you were afraid of him?
Should you be scared of his phalangites? But you have plenty of heavy infantry of your own. The Assyrians and Egyptians have served you well in the past; the latter wield their own pikes with skill, while hiding behind their heavy tower shields. You have Lydians and Phrygians and Lykians and Phoenicians too. The Persian elite infantry itself gives a good account of itself in close fighting. But if all these are not enough, you have money to buy Greeks, if they are so keen to fight. That king Alexander fields a paltry 9,000 pikemen; send out your envoys to the Peloponnese and you can gather 20,000 or even 30,000 hoplites to confront them. And that's to say nothing of your scythed chariots, ready to carve up the enemy formations.
Should you be scared of his heavy cavalry then? His vaunted Companions? But you have royal Companions of your own, numbering 4 times the Macedonian total, heavily armoured and happy to fight in close combat to guard your person. And then there's all the others - Medes and Baktrians and Sakai and horsemen from all the other peoples who live in your empire. Why should you be worried about a few of Alexander's drinking buddies led by his favourite boy toy?
The fact is that Alexander invaded with an army that could not match Persia's potential and that had, on the face of it, no notable edge in pitched battle. There was no reason for Darius to think he would struggle to beat him.
3. It was beneath the king's dignity to run
Persian royal ideology laid a heavy emphasis on the Great King's role as a general and protector of his people. Both Greek and Persian sources show the pressure on the ruler of Persia to show himself capable of leading an army and vanquishing the empire's enemies. This was what legitimated his rule: his divine mandate to protect his subjects from violence and usurpation.
It simply would not do to run away from this arrogant Macedonian. The empire was under attack, and the Great King's sole purpose now was to take direct and effective measures to crush its enemies. For mere troublemakers on the border it was fine to delegate the defence to local satraps, but once Alexander had moved further into Asia Minor, it was clear that Darius should act. Had he failed to lead his own army his reputation would have taken a serious blow, and he may have found it difficult to raise either money or troops to continue the war. Indeed, if he was regarded as weak he would have to fear intrigue and assassination from relatives who thought they could do better.
Far better, therefore, to gather the empire's troops in a single indomitable force and bring it down on the invaders like a ton of bricks. While Darius was known as a physically imposing fighter, it wasn't really necessary for him to engage in hand-to-hand combat, as long as he was seen leading the army of his empire to victory. And as noted above, he had every reason to believe that he could do so.
4. There was no proper alternative strategy
Even if Darius had reasons to avoid battle, and even if he could somehow get away with doing so, it wouldn't have been a clever move, since there was little else he could do to defeat the invading army. If the Persian Empire had consisted of only deserts he might have been able to starve Alexander out; after all, Alexander would almost do this to himself some years later, without even an enemy to goad him into it. But the lands of the Empire's western half were largely fertile and wealthy, and could easily have sustained Alexander's campaign. If Darius had abandoned them, he would only have strengthened Alexander.
Fabian tactics, after all, are based on the notion that the invading army isn't attempting to set up shop, but only to defeat its enemy. Alexander showed from the outset that his intention was to replace Persian hegemony with his own. When he conquered cities and regions, he would impose tribute, and resume the regime of imperial exploitation that the Persians had introduced. Therefore, avoiding battle and leaving him to stomp around the countryside would result in nothing but a steady trickle of resources from Persian coffers to Macedonian ones. How exactly was this going to beat Alexander?
Ancient armies usually couldn't be kept in the field for very long, because they were either militias (consisting of people who needed to get back to their farms) or professional soldiers (who consumed vast quantities of cash). Alexander's army mostly consisted of the latter. But the huge stockpiles of bullion he "liberated" in Sardis and other places meant that he woud be able to keep that army going for as long as it took, and the inflow of tribute meant that the money would never run out. In this context, to avoid battle was to condemn the Persian Empire to a death of a thousand cuts. It would be slowly but inexorably conquered as the balance of power swung toward the Macedonians. Darius could not afford to let this go on; and, as noted above, he had no reason to let it go on and faced tremendous pressure to aim for a big showdown. So a showdown is what he engineered for himself, and a showdown is what he got.