There were a few reasons for Muslim expansion on both the Persian and Byzantine fronts, the sum of which boils down to good ol' Realpolitik and Sphere of Influence doctrines. And I'm going to assume that you have at least a superficial understanding of the rise of Islam and Mohammad's attempt to unify the Arabs of the Arabian Peninsula and bring them into the Islamic fold.

Well, there were two major Arab tribes at the northern ends of Arabia, the Lakhmids and the Ghassanids. The former were vassals of the Persians and the latter were vassals of the Byzantines. Both tribes were utilized by the great powers in their wars against each other, as they were excellent complimentary auxiliary forces to their respective armies when they clashed, and were able to continuously raid the other's lands and harass enemy forces, then disperse and melt back into the desert.

But another role they played, one that's more pertinent to your question, is "buffer states". They initially held back the other disparate tribes from the Sassanid/Byzantine borders by virtue of being the powerful, connected players holding the borderlands.

Enter Mohammad. The Ghassanids and Lakhmids were hostile to Mohammad and his message, as his rise would naturally come at their expense, and were spurred on by both great powers, who wished (naturally) to suppress this rising power to their south. They helped both these tribes against Mohammad's federation and were overall hostile to it. But Mohammad eventually prevailed, scattering the two tribes and absorbing the remnants into his nation, and in effect, imposing himself upon both empire's Spheres of Influence.

As you can see, the seeds of mutual enmity were already planted, the Causus Belli already set in the forms of encroachment by Mohammad, and hostility against this rising upstart who had cleaved away both great power's vassals and stood at their borders, but Mohammad didn't invade just yet. He sent emissaries to both empires, inviting their leaders into his faith, and much more importantly, asking for permission to propagate this new faith in both empire's lands in return for peace, and of course, he was rejected on both counts.

The seed took root. Mohammad knew that he would always be perceived as a threat to the powers, who might (and did) eventually see common cause in his nascent nation's destruction, and he had already sent his emissaries, who were rejected. What was left for him but war?

The seed eventually flowered. The first and second Caliphs send out armies of motivated men, lead by brilliant commanders, who were fighting militarily exhausted, politically corrupt, and financially bankrupt husks of former glory. And the rest, as they say, is history...