The Athanasian Creed in General
The Athanasian Creed is so called because it was traditionally written by St. Athanasius, notable for his defense of the trinitarian doctrine of God against the novelty of the priest Arius. However false the attribution to Athanasius, the idea that it does not constitute a faithful exposition of the Catholic faith as he saw it ha sno merit whatsoever—it is clearly a distilled statement of his faith. It is, if not Athanasius' Creed, a markedly Athanasian one, in other words.
Said credal formula involves the most basic doctrines of "the Catholic faith,... without keeping which whole and inviolate a man will without a doubt perish eternally." Like the Nicene Creed, its focus is on the doctrine of God in particular, and specifically the triune nature of God. But whereas the Nicene Creed is content to affirm at an Ecumenical and thus dogmatic Council of the Church that the Son is divine in the sense of sharing the nature of the Father, not merely partaking in some way in it, the Athanasian Creed focuses on the precise details of the triune nature of God, such as the absolute equality of the persons, so that it is clear whether you hold to an orthodox doctrine of the Trinity, or have a modified 'Trinity lite.'
The Nicene Creed also has "God of God, Light of Light, true God of true God," which clause is intended to reject the hersey of Arianism which does not believe that the Son is true God, but a creature, and affirm that the Son shares the nature of God because He was "begotten of the Father before all ages." You won't find the deity of the Holy Ghost in the Nicene Creed because the Arian heresy was one which denied the divinity of the Son; it was not a Pneumatological heresy (or heresy regarding the Holy Ghost)—which would be addressed in a later Council, Constantinople I (and likewise, other Christological doctrines not delved into at Nicaea).
The Unity of God in Scripture
Scripture leaves no room for us to consider the persons of the trinity to be other than the one God—and moreover affirms that they are. In case the "three" and "one" in the Trinity are being conflated or made univocal, it should be first understood that the "one" refers to one aspect of God's nature, and "three" another—the same aspect is not both one and three at the same time, which is nonsense-speak, not a mystery: the mystery is how a nature can be equally said to belong to three persons, and yet be one nature. This is how it is possible to speak of one God, but three persons. However, this is at least not unthinkable when considering that the divine nature is fundamentally incomprehensible to man anyway—if the name Michael means "Who is/can be like God?" the one could ask also, "what is like God?" to which the answer to said rhetorical questions is, "no one, and nothing." And if nothing is like God, we can at best understand God's nature through imperfect analogy, and certainly not as He is: "no one can see my face and live" (Exodus 33:20). God is invisible; here, to see means to experience in an unmediated way (expressed by analogy to sight. We digress.
One God, the Father
Trinitarians of course, as you note, affirm the doctrine that the one God is the father. However, this doesn't preclude the Trinity any more than John 17:3 (DRB):
Now this is eternal life: That they may know thee, the only true God, and Jesus Christ, whom thou hast sent.
If the Father is not the only true God, then the Trinity is false by definition, because it asserts Him to be such. Who is He supposed to be? It should also be noted that "true God" and "God" are synonymous here, since "the God" is "the true God." Therefore, this verse is really only saying the Father is God, and other false idols and gods are "nothing in the world, and there is no God but one" (1 Corinthians 8:4). It does not help the Unitarian as he thinks it does.
What non-Trinitarians do is reword this verse from "you are the only true God" to "only you are the true God," making the verse not about what the Father is, to what the Father exclusively is, which is clearly Semantically different—and only equivalent to the actual verse if it is assumed first that God must be unipersonal, rather than tripersonal. If "the divinity of the Father" is seriously taken for an argument against the Trinity, then the Trinity has not been apprehended properly. Just as someone who says, "God is not a man" to the Incarnation is ignorant that such is a premise necessary for the Incarnation to be true.
Trinitarians also believe that the Son is the true God, and the Holy Ghost is the true God (precisely because there is only one divine nature—one God; anyone believed to be God, namely the Trinity, is considered to be synonymous with that divine nature, not a 'one-third participant'). Note that the Creed of Nicaea, which asserts the equality of the Son with the Father, and that they share and are one substance (or nature) says that the Father is the one God we believe in; it is simply that by extension, anyone who shares His nature or substance is just as much the one true God by definition. Thus the Creeds do not have a different faith; the Athanasian is simply a far more in depth and precise look at relationship of the persons.
The Scriptures cited by the other answerers are sufficient, so I don't want to belabour the point by re-citing them. Besides, the Scriptural 'proof' for both Unitarianism and Trinitarianism is always debatable by someone. We don't rely on Scripture minus the faith which grounds them: as Athanasius said, it becomes necessary to cite them as proofs only insofar as they are rejected by heretics who ignore tradition and assign the Scriptures a novel interpretation, and not as though the doctrine stood or fell on someone's acceptance of a Trinitarian explanation of them. In other words, to vindicate the Scriptures' meaning, "that they bear the orthodox sense."
TL;DR
Since the persons of the Trinity have one nature or substance, not three separate natures, they are one of that substance (God). This is why Jesus will be able to say, "I and the Father are one [God]," or that "He that has seen me has seen the Father." Not that they are identical personally, but both are identical to the divine nature.
