This is a tricky question that very easily gets mired down in details and definitions and there are a lot of angles to approach it from. Explicit statements of monotheism are first seen in the so-called Deutero-Isaiah, which was written in the post-Exilic period (that is, the 5th century BC or so). It lagely consists of formulaic praise of YHWH as a creator deity along with repetative phrases along the lines of "there is only one God".

So, what's important to understand is that it is strictly speaking only in this period that one should speak of "Judaism" and "Jews" as opposed to "Israelites" or "Ancient Hebrews", which means that Judaism is almost definitionally monotheistic. But that's a bit of a nitpick - what are the origins of the Jewish idea of God?

If you read the Hebrew texts, you will find two words used to refer to God - "YHWH" (also written Yahweh, rendered the LORD in English translations, based on the Masoretic practice of reading the holy name as "Adonai", literally meaning "Master of the house") and "El"/"Elohim" ("Elohim" is a superlative along the lines of, "the most exalted El"; El also occurs with epithets and is translated "God" in English Bibles). "El" is also a generic word for god, and it ultimately derives from a Near Eastern deity who featured as a king of the gods in early Levantine beliefs (whom the Greeks for example identified with Kronos, the father of Zeus.) This is actually preserved in Psalm 82, which puns "Elohim judges among the elohim", that is, among other gods.

YHWH is nowhere to be found outside the Hebrew Bible (EDIT: As someone pointed out to me, there is the Mesha stele, which refers to "vessels of YHWH" in Israel several hundred years before the Tanakh was compiled. However, this is a rather moot point since it's around when the oldest oral traditions are dated to (8C-9C BC), refers to Israel, and contains no useful information about the deity. Hence it changes little.) but there are strong parallells with the Ugaritic storm deity Ba'al in Ugaritic texts, who also has some overlap with El (they are both symbolized by bulls, for example). YHWH is said to have crushed the head of the sea serpent Leviathan in Psalm 104, and he boasts of a similar feat in the climactic ending of Job; similarly, Ba'al slew the sea serpent Lotan. There are a few other associations between YHWH and storms, such as the fact that he speaks from a whirlwind in Job and parts the red sea with a mighty wind.

Now El had a consort, Asherah, of whom figurines have been discovered in relatively old layers preceding the aforementioned exile, suggesting that at some point they were discouraged or outright forbidden as unlawful idolatry. Meanwhile, Yahweh often appears as a wrathful deity, an avenger who metes out harsh justice. The Babylonian Exile was considered a period of enormous suffering and humiliation for the people of Israel, and in such a time it is understandable that they would crack down on things that might seem impious (idols) and turn increasingly to the worship of a wrathful avenger deity. Themes of injustice are central to the writings of the actual Isaiah (who predated the exile), which is probably the reason for deutero-Isaiah being attributed to him.

There is no entirely satisfactory explanation for how the ancient Hebrews (or some of them, at least) went from exclusive worship of Yahweh to the very strong statements made in deutero-Isaiah. The most commonly (and reasonably) suggested influence is the Iranian religion, Zoroastrianism, which features a supreme being in the form of the uncreated creator Ahura Mazda. I go into Zoroastrian theology at length here and its origins here. At the very least, Zoroastrianism and Judaism share a lot of ritual features (particularly a focus on ritual purity, apprehension towards idolatry, and ethnic exclusiveness) which may have made the Israelites particularly sympathetic to them. Indeed, Deutero-Isaiah spends a great deal of time lauding the Great King Cyrus as guided by YHWH, and the statements exalting YHWH as creator deity bear a striking resemblance to invocations of Ahura Mazda in both contemporaneous Old Persian inscriptions and older Zoroastrian scripture. The Prophet Zoroaster (Zarathustra) was a figure of great antiquity even in those days, and judging by Greek writers of the time, many (e.g. Plato) considered him a venerable source of ancient mystic wisdom and magic, so it is certainly conceivable that some concepts could have been transmitted. Another plausible influence is the shift of understanding Satan as a servant of God into the epitome of evil we are familiar with today. I'm currently working on trying to make a more or less exhaustive detailed list of possible influences and parallels, but it's still a work in progress in the early research stage.

Regardless of how great we take the Zoroastrian influence to have been, it cannot completely explain the strict focus on God's one-ness and repeated assertions that there are no other gods, etcetera, because these aren't exactly features of ancient Zoroastrian theology (which is however very complex and difficult to truly comprehend). The full explanation, then, appears to be lost to recorded history.