The actual existence of God is implicit in, and so necessitated by, the very concept of God. This is what Aquinas is getting at in his argument that the actual existence of God is essential to his nature.
That is not at all what St. Thomas Aquinas was 'getting at' in that argument, an which is less his than Avicenna's; what you get in Aquinas is a summary of what is fully spelled out in Avicenna's Metaphysics, which is the thirteenth book of his Healing.*
What these philosophers meant is to elaborate a point of Aristotle's, which defines goodness in terms of desire. Aristotle says that the good is what is desirable, and for the most part that differs based on what the thing is doing the desiring. Rain may be very desirable from the perspective of the grass, and therefore it is good for the grass; it may be less good for the sailors at sea, because it is less desirable. If we want to talk about The Good per se, then, we would need to find something that is desirable for all things. Everything, Aristotle suggests, desires existence: both grass and sailors, squirrels and trees all seek to preserve their existence and to extend it through reproduction. In the Metaphysics, the great spirits that drive the stars seek to imitate the perfection of the Unmoved Mover, which they are able to do only by rough imitation: they travel in circles (as the Greeks believed they did, based on empirical observation) because the circle has a kind of infinity in its eternal surface that never ends.
Avicenna goes on to point out that existence is, then, goodness per se: "to exist" and "to be good" are thus phrases that differ only in emphasis. (Aquinas gives his summary of this part in ST I 5a1.) Of these two concepts, being has priority (next article in Aquinas); and thus the idea of goodness derives from existence.
But God's existence is necessary -- and here is where Avicenna greatly outpaces Aquinas' summary. Avicenna demonstrates with two long form arguments that the universe would not exist if it were not for the existence of another being that exists necessarily: and, having proven that such a being must exist, he also proves through another set of arguments that there can be only one of them. Thus, the Orthosphere is wrong to suggest that Aquinas is merely suggesting that necessity is 'contained in the concept' of God. The proof Aquinas is citing is a proof of the logical necessity of exactly one being whose existence is necessary. It is not contained in the concept, but separately established.
Aquinas' summary of this is in ST I 2a3. He gives summaries of five arguments for God's existence. The first one is from Aristotle's Metaphysics, the argument for unmoved movers. He then gives summaries of both of Avicenna's arguments.
The second way [to prove God's existence] is from the nature of the efficient cause. In the world of sense we find there is an order of efficient causes. There is no case known (neither is it, indeed, possible) in which a thing is found to be the efficient cause of itself; for so it would be prior to itself, which is impossible. Now in efficient causes it is not possible to go on to infinity, because in all efficient causes following in order, the first is the cause of the intermediate cause, and the intermediate is the cause of the ultimate cause, whether the intermediate cause be several, or only one. Now to take away the cause is to take away the effect. Therefore, if there be no first cause among efficient causes, there will be no ultimate, nor any intermediate cause. But if in efficient causes it is possible to go on to infinity, there will be no first efficient cause, neither will there be an ultimate effect, nor any intermediate efficient causes; all of which is plainly false. Therefore it is necessary to admit a first efficient cause, to which everyone gives the name of God.The third way is taken from possibility and necessity, and runs thus. We find in nature things that are possible to be and not to be, since they are found to be generated, and to corrupt, and consequently, they are possible to be and not to be. But it is impossible for these always to exist, for that which is possible not to be at some time is not. Therefore, if everything is possible not to be, then at one time there could have been nothing in existence. Now if this were true, even now there would be nothing in existence, because that which does not exist only begins to exist by something already existing. Therefore, if at one time nothing was in existence, it would have been impossible for anything to have begun to exist; and thus even now nothing would be in existence — which is absurd. Therefore, not all beings are merely possible, but there must exist something the existence of which is necessary. But every necessary thing either has its necessity caused by another, or not. Now it is impossible to go on to infinity in necessary things which have their necessity caused by another, as has been already proved in regard to efficient causes. Therefore we cannot but postulate the existence of some being having of itself its own necessity, and not receiving it from another, but rather causing in others their necessity. This all men speak of as God.
God's existence is necessary, then; and his goodness is then both not only also necessary, but of a kind that is just as different from ours as our possible and temporary existence is different from God's necessary and eternal one.
His fourth and fifth proofs are also Aristotelian -- he explicitly cites Aristotle for the fourth one. His invocations of Anslem are generally critical, though as he was criticizing a canonized saint while he was yet a living man he had to couch his criticisms in respectful terms. Still, Anslem was not what he was 'getting at' in his work.
* Usually "The Healing," because Arabic is one of those languages that prefixes nouns with a definite article, as French does "la" or "le," and Spanish does "el" or "la." Just as "La France" is really "France" in English, though, the definite article is not really always necessary or appropriate in translation.
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