The short answer is that when they were moving south from Central Asia (from approximately the region of Kazakhstan and Uzbekistan), through Iran, they turned right (west) and ended up in upper Mesopotamia. Those who brought Sanskrit to India turned left (east) and ended up in India. This movement of peoples and cultures and languages was part of the Indo-European expansion, which had branches to the west, south, and east. The southern branch was the Indo-Iranian part of the Indo-European expansion.
The longer answer is that it isn't certain that there were actually Sanskrit speakers in Syria in 1500BC. The people in question are the Mitanni, and the Sanskrit connection is:
The Mitanni aristocracy often had Indo-Iranian names (or we can say early Sanskrit names), at least as reign-names, even if they had non-Indo-Iranian names before then.
They had gods with Sanskrit names - Varuna, Mitra, Indra and Nasatya.
They used Indo-Iranian vocabulary for chariotry and horse-training. At least, a Hittite text attributed to "Kikkuli, master horse trainer of the land of Mitanni" (and "master horse trainer" is assussanni, cognate with Vedic Sanskrit aśva-sana) used Indo-Iranian terminology.
The problem with saying that the Mitanni spoke Sanskrit is that by the time we have enough sources to say what they were speaking, they were speaking Hurrian (which is not Indo-Iranian). Hurrian with Sanskrit loanwords for chariotry and horse stuff is not Sanskrit (or Indo-Iranian). Clearly, there was contact and influence, but we don't know that the Mitanni ever spoke an Indo-Iranian language.
Anthony (2007) speculates that the Indo-Iranian influence was from mercenaries who became rulers:
A good guess is that the Mitanni kingdom was founded by Old Indic-speaking mercenaries, perhaps charioteers, who regularly recited the kinds of hymns and prayers that were collected at about the same time far to the east by the compilers of the Rig Veda. Hired by a Hurrian king about 1500 BCE, they usurped his throne and founded a dynasty, a very common pattern in Near Eastern and Iranian dynastic histories. The dynasty quickly became Hurrian in almost every sense but clung to a tradition of using Old Indic royal names, some Vedic deity names, and Old Indic technical terms related to chariotry long after its founders faded into history. This is, of course, a guess
But this isn't the only reasonable guess. The chariotry vocabulary could have been adopted along with the chariot, and names of gods and kings can be adopted along with religion. Anthony's guess that Indo-Iranian charioteers imposed themselves as rulers (whether mercenaries-turned-conquerors or simply conquerors) is reasonable, but if it is the case, we don't know when they switched from speaking Indo-Iranian languages to speaking Hurrian (adopted from the ruled people). Possibly, the change was before we can speak of the Mitanni.
The connection with chariots is that two key elements of Indo-European culture spread west, south, and east in the Indo-European expansions: Indo-European languages and chariots. The spoked chariot wheel appears to have been invented around 2000BC, and may have enabled a large part of the Indo-European expansion. The chariot appears in the Andronovo culture, probably Indo-Iranian or Indo-European speakers (mostly), with the earliest finds in the purple region in this map:
The combined red-purple-orange region is the extent of the Andronovo culture, and the origin of the southern Indo-European expansion (i.e., the Indo-Iranian expansion). Given that the languages that gave rise to Sanskrit had to move south from this area to where Sanskrit and its descendants are spoken today, it's no surprise to find languages closely related to Sanskrit along the way, or close to along the way - such as in Upper Mesopotamia.
Reference:
David W. Anthony, The Horse, the Wheel, and Language: How Bronze-Age Riders from the Eurasian Steppes Shaped the Modern World, Princeton University Press, 2007.
