Could you explain this a little more? I am unsure what you mean by entering the life of the holy trinity.
That is quite a tall order! I will do my best, but please remember that asking an Orthodox what communion is can be like asking a Buddhist what enlightenment is, so the answer may take time or go beyond the scope of this conversation. In fact, I am almost tempted to say that this question cannot be answered but only experienced as reality.
I will define anything I say that may not be familiar to a non-Christian, not in an attempt to be condescending, but to avoid confusion.
In the Old Testament(an early book of sacred writings held in common by Jews, Christians, and Muslims) when God reveals Himself to Moses on Mt. Sinai, Moses asks God to share His Name. God responds, "I am that I am" or more simply, "I am He who Is." We believe that there is one God who exists in three Persons; God the Father, God the Son(Jesus Christ, also known as God the Word), and God the Holy Spirit(hereafter referred to as The Holy Trinity). It is impossible for us to define God, but it might be said that God is love that the Holy Trinity can be described in relation to each other as: The Lover, the Beloved, and Love itself.
Human beings were created in the image and likeness of God so that they might become part of this divine relationship. But as true love cannot be forced, only compelled, God created human beings with the freedom to accept or reject this relationship. As human beings were created to exist in this state of communion with God, rejecting this is also rejecting what it means to be human. The first humans rejected this communion with God, and as a result death entered into the world, not as a punishment by God, but as the effect of the cause, just as if you cut yourself you will bleed, not as punishment from the body but as because that is how bodies are made to sustain themselves. This is referred to by Christians as the "fall of mankind." We teach that as death entered into the world, it became part of living creatures, passing death down to their descendants, as parents pass hair color and other physical traits.
At this point human beings were cut off from this relationship, the very essense of their being. But how can God correct this without violating the free will He gave humanity? To restore human beings to their essense as creatures in communion with God, God took on human flesh in the person of Jesus Christ. Jesus Christ was 100% God and 100% man. Jesus Christ entered the world in flesh roughly two thousand years ago in the lands occupied to the descendants of the people in the Old Testament. As He was God, the source of infinite mercy and love, He forgave others of the wrongs they have commited. This claim to be God caused the religious authority of that time and place to declare that he was a blashphemer and brought Him before the goverment authorities and was executed on a cross. Christ did not oppose these people as His death was to serve an even greater purpose. As God who is life died, death itself died as it could not contain the life of God, and through this voluntary sacrifice of Christ restored the ability of human beings to be in communion with God and with each other.
When I speak of entering into the life of the Holy Trinity, it is this: God became man so that man might become like God. By God's grace we are able to enter into the exact life that God is as the love of the Holy Trinity the way that Trinity by nature is this life.