This explanation could probably use some work, but the eucharistic bread is consecrated to become Jesus’ body. However, since Jesus’ person is indivisible, his blood, soul, and divinity are also made present under the appearance of bread. Likewise, the wine is consecrated to become his blood, but as Jesus is indivisible, his body, soul and divinity become present there as well.
I imagine the explanation you heard has similar logic (if not exactly the same). It is Jesus specifically who is being made present in his fullness in the Eucharist, and in addition to his human body, blood, and soul, there is also the divinity. But the Son is not divisible from Father or Holy Spirit, so the entire Trinity becomes present in the Eucharist by its association with Jesus who is made present.
So the Trinity is present, the Son primarily and the Father and Ghost therefore by their indivisible association with the Son.
But to keep it very plain, the Trinity is present. How Father, Son, and Holy Spirit are present, if you really wish to dig into it, might require a more complicated explanation, but all three are there.