It is as complex as God:
Just as Theists are unable to give logical explanation to Atheists about the existence of God, so is the case with Christians, where it is impractical to give a logical explanation of Triune nature of God because that which is being attempted to explain (in both instances) refer to same Entity called God. The concept of Trinity is as enigmatic as the existence of God and it is obviously so, because ultimately we are discussing about the same God.
The "Trinity" is an expression used to articulate one of many attributes of one and only one God based on the various references found woven through all of Scripture from Genesis to Revelation, which is why the doctrine was formulated. There is equally clearly only one true God in the Bible. Numerous references are available for both. It is simple logic to end up with the Trinity, although how it works in practice must remain to a degree a mystery, since if we fully understood God in His entirety, He would not be God.
The search for proof of existence of God is inconceivable in our finite mind and beyond the ability of limited conception of human intelligence & imagination. With this presumption, we get our belief in God and same is true for Trinity since with Trinity we are trying to define same God. In other words just like you cannot define God by your logical and scientific mind, the same is true of Trinity because we are trying to define the same God.
You and I are individual personalities, just like every other god of the major religions. The idea of the Trinity has plurality Within unity, a composite unity. This isn’t saying there is more than one God.
Way out for accepting it:
For Atheist it is the non-belief in (existence) God that keeps him from becoming a theist. For theist it is the “constituents, frame and state” of mind, not the proof, which makes him accept the belief in existence of God. However, in some cases with a presupposed mind, like a person with Islamic background, it does prevent them from accepting the Christian faith because of his inability to accept Trinity in the same “constituents, frame and state” of mind as he is accepting the existence of God.
Trinity is woven in scriptures:
Some argues that when it is not specifically stated anywhere in the Bible about the triune nature of God then why to formulate a Doctrine for it? Why not live and practice Christianity without it?
The answer lies in the truth that one cannot be a follower of Jesus if you do not believe in what He said and preached. When Jesus came into this world and preached the Good News about God’s kingdom, the true nature of One Almighty God was also revealed. Most obvious and direct evidence for Trinity emanated from Bible, depicting Jesus Christ and Holy Spirit with Nature of God. Therefore, if Father is God, Son is God and Holy Spirit is God than that’s it. Bible gives us irrefutable references about these claims in amorphous manner in OT and explicitly in NT. It was amorphous in OT, the reason being that it never became that clear until the Son was revealed in NT.
The central issue is, do we believe the Word of God? Despite what many non-Christians believe, Christians – (Mainstreams Christians) – believe that there is only one God. Yes, the Trinity is an odd doctrine, and many don’t like it. But who would make it up? We humans like things to make sense. Only the Trinity is an idea so odd that even after two millennia we still can’t properly understand it.
That there is one God, and yet, someone, there are three personalities who are deity, and yet there aren’t three gods but one. It sounds weird, crazy even. Does it seem that humans would make something like that up, or that the devil would? Satan knows that we like things to make sense; he wouldn’t create such a puzzling doctrine, neither would we humans and St. Constantine included.