How did the political conflicts of the Persian and Greek empires contribute to an exchange of cultures ?

First, one might want to rethink the phrasing of this question. The main area of cultural contact between Greece and Persia was Asia Minor, where Greek city states were in geographical proximity to Persia and on major trade routes. Persians visited these city states and Greeks visited Persia leading to substantial cultural interchanges, especially in science, religion, medicine, the arts, and technology. This degree of contact was enhanced by peace rather than conflict, and the Persian Wars really were an interruption to this. Also, you should note that Greece was not a unified empire, but a group of city-states with constantly changing allegiances. Athens, briefly, tried to transform the Delian League into something approximating an empire, but this was short-lived and generally unsuccessful.
A far greater impetus to cultural exchange was internal conflict in Greece, especially the Peloponnesian wars. During these, Greek factions would ally themselves to Persia in an attempt to gain support against other Greek city states. Many Greek soldiers served as mercenaries in Persian armies, leading to increased cultural interchange in the late fifth and fourth centuries. Xenophon provides us with an account of serving as a mercenary for Persia and is one of our best sources on such cultural interchanges.

The Persian and Greek political conflicts occurred between the Achaemenid Empire and the Greek city-states around 499 BC. These two political powers represent conflicting philosophies in governance, but hold many similarities in regards to cultural developments. 
During this period in the Middle East, the Persian Achaemenid Empire was sweeping force that unified large territory under one central government. This led to the building of large infrastructure  such as roads, an official language and an overall exchange of goods. All of these aspects allow for the interchange of ideas and cultural mores over a territory that had been divided. 
In contrast the Greek city-states represented independent political systems, and it was with the influence of outside sources and threats ie. Persian conquest that led them to unify. They too were interested in expansion and held colonies in a variety of areas, including the Middle East, Southern France and Italy. They had established large trade naval routes between their colonies. 
The political conflicts, and the desire to expand and conquer brought these two systems to butt-heads, however there had already existed an exchange of culture between these two entities. Many objects and that were created and development from Persian territory had made there way through trade to the Greek city-states. This can be seen in religious beliefs, luxury goods and technology found in Greece and Etruria. 
These conflicts made the exchange of culture more substantial through direct involvement and interaction between the Greeks and Persians, as well as leading them towards more development in infrastructure and naval development which led to stronger trade ties. Luxury goods were seen as exotic and purchased by the elites in both cultures, this would eventually trickle down and spread through both cultures.