The first major battle was Tel Buwayb in 635, breaking into Sassanid territory. In 636 comes Ghadasia, in southern Mesopotamia. In 637 comes Al-Qadisiya near present-day Baghdad, and in the same year the defeat of the Byzantine army at the battle of Yarmuk, which leads to the fall of Damascus. In 638 Jerusalem is beseiged and surrenders. In 639 the conquest of the Sassanid empire continues, with the fall of Edessa and Haran. In 641, the Byzantines are driven out of Alexandria, and in 642 Cyrus, the Alexandrine patriarch, surrenders all of Egypt and pays tribute in return for a promise of toleration. Also in 642, off in the heartland of the Sassanid empire (modern Iran), the Muslims win the battle of Nehawad. In just seven years, Omar leads them from a modest Arabian state to the dominant power in the near east. Two years later he is dead himself, assassinated in Medina by a slave of the Muslim provincial governor of Syria, a figure in the Omayyad house that a generation later founds the dynasty.
The Sassanid empire had been weakened before all of this by its defeat by the Byzantines in the battle of Ninevah in 627. That was a reversal, following up the reconquest of Syria and Palestine, which the Sassanids took just before, in an earlier campaign, still in the 600s, that brought them to the outskirts of Constantinople before being driven back. So what you have to understand is the area conquered was in a whiplash between east and west, with now the Sassanids, now the Byzantines, blitzing the same general area. Omar came in first against the temporarily weaker party, the Sassanids, gained territory by doing so, and turned on the other party within 2 years of setting out, beating both, conquering the temporarily weaker party outright, and tearing away large provinces from the other in a very short space of time.
There is no way such events can be hidden, minimized, obfuscated, or recast. The Sassanid empire was flourishing two decades earlier and threatening Constantinople, and in the blink of an eye it is just gone, caput, nada, disappears from history. Byzantine rulers were campaigning in central Iraq and Byzantine theologians were worrying over enforcing orthodoxy against Nestorian and monophysite heresies in Syria and Egypt one minute, and the next minute they have been ejected from the near east entirely, never to return.
Understand, the -early- period of Islam, the time of Muhammad himself, is obscure in details. The reason is nobody else outside gave a hoot before it turned into something so big (reason one), and all the stories from the Arabs themselves have been embellished and recast for their own later motives of legitimacy, to make such-and-such legal by some precedent for it in the time of Muhammad. But the basic fact that an armed monotheism rode out of the Arabian desert and conquered the near east like lightning, is not disputable. It upsets every existing historical process in the whole area.
Origins are often shrouded in mystery, since the importance of seminal events generally cannot recognized at the time, but only in retrospect.