First of all, the dogma of the Trinity is a Mystery of our faith, which means it cannot be fully comprehended. However, it does not mean that it cannot be understood at all. But just like we can never fully know everything about another person, we are well capable of understanding something of one another and growing in the depth of that knowledge over time. It is the same with the Trinity.
Perhaps the key to begin to understand how God can be one and yet three Persons is that our God is a relational God. God reveals Himself to us as a relational God: Two Persons intensely give of Himself to one another that the Two become inseparably One and that love overflows and becomes a Third Person, also fully in union with the Two. God is love and this is pure love: selfless giving, selfless receiving, total union, selfless overflowing outward…
And thinking of that, I could not help but realize that this has always been what I long for as well: to give of myself freely, to be loved openly, and to bear fruit by enabling others to experience that love, and together draw into a deeper and deeper union and communion. I believe that’s the longing of every heart. This is what truly sets the human heart free. And that is the reality and proof that we are made in the image of the Triune God.
Of course, the above imagery is only an analogy: God does not “become” as in changing from one moment three separate entities and later by love uniting into one. That final reality of love has always been the only reality of God from before time began to beyond time’s expiration.
One can also argue that a non-Triune God necessarily cannot be a self-sustained relational fulfillment, meaning such a God cannot be relational and fulfilled without engaging someone else outside of Himself, in which case creation would become a necessity partly to fulfill His need to love, instead of a pure gratuitous gift. But this will go against everything God has revealed to us about Himself in Scripture and through Jesus.
The above, of course, is not a proof, but certainly worth pondering. Just like any relational truth, it cannot be proven, but can only be grasped with the heart, which is often keener than our mind.
Lastly, if you find yourself in a situation that you need to explain the Trinity, it would be best to speak from your own experience of His Triune love. Evangelization is rarely achieved through eloquent philosophical arguments, but by attraction through personal conviction and testimony. We preach not to convince, but to prove that we are convinced, because we have experienced the Triune love of God.
This love is for you, too. Do you want to experience it as well? Let us find it together…