In what way is the Son of God "one with the Father"?

The Trinitarian position on this is pretty straight forward. Hard to wrap your mind around, maybe. A mystery, sure. But the same basic idea is held by most Trinitarians, especially Protestant and Catholic traditions. Eastern Orthodoxy comes at this from a rather different angle, but as far as application to that verse goes aren't fundamentally different.

Jesus is one substance with God the Father. In other words he was and is God incarnate, not some created being, not a lesser creature, not a mirage, really and truly the one true God in physical form.

The Athanasian Creed is a good reference for this point:

Now this is the catholic faith:

Note the little-c catholic here is not a reference to Roman Catholicism, it just means universal.

That we worship one God in trinity and the trinity in unity,
neither blending their persons
nor dividing their essence.
For the person of the Father is a distinct person,
the person of the Son is another,
and that of the Holy Spirit still another.
But the divinity of the Father, Son, and Holy Spirit is one,
their glory equal, their majesty coeternal.

That's the basic outline of what the Trinity is: three persons but one being.

Now here's the part that directly addresses that verse. It says basically the same thing over in a lot of ways just to eliminate as much confusion as possible and not leave wiggle room for novel definitions:

What quality the Father has, the Son has, and the Holy Spirit has.
The Father is uncreated,
the Son is uncreated,
the Holy Spirit is uncreated.

The Father is immeasurable,
the Son is immeasurable,
the Holy Spirit is immeasurable.

The Father is eternal,
the Son is eternal,
the Holy Spirit is eternal.

And yet there are not three eternal beings;
there is but one eternal being.
So too there are not three uncreated or immeasurable beings;
there is but one uncreated and immeasurable being.

Similarly, the Father is almighty,
the Son is almighty,
the Holy Spirit is almighty.
Yet there are not three almighty beings;
there is but one almighty being.

Thus the Father is God,
the Son is God,
the Holy Spirit is God.
Yet there are not three gods;
there is but one God.

Thus the Father is Lord,
the Son is Lord,
the Holy Spirit is Lord.
Yet there are not three lords;
there is but one Lord.

Just as Christian truth compels us
to confess each person individually
as both God and Lord,
so catholic religion forbids us
to say that there are three gods or lords.

The Father was neither made nor created nor begotten from anyone.
The Son was neither made nor created;
he was begotten from the Father alone.
The Holy Spirit was neither made nor created nor begotten;
he proceeds from the Father and the Son.

Accordingly there is one Father, not three fathers;
there is one Son, not three sons;
there is one Holy Spirit, not three holy spirits.

Nothing in this trinity is before or after,
nothing is greater or smaller;
in their entirety the three persons
are coeternal and coequal with each other.

A couple paragraphs later it gets into more specifically how this relates to the Son's incarnation. Obviously him taking on human nature adds an extra wrinkle to the whole thing, so it points out that this doesn't nullify the basic Trinitarian formula:

That we believe and confess
that our Lord Jesus Christ, God's Son,
is both God and human, equally.

He is God from the essence of the Father,
begotten before time;
and he is human from the essence of his mother,
born in time;
completely God, completely human,
with a rational soul and human flesh;
equal to the Father as regards divinity,
less than the Father as regards humanity.

Although he is God and human,
yet Christ is not two, but one.
He is one, however,
not by his divinity being turned into flesh,
but by God's taking humanity to himself.
He is one,
certainly not by the blending of his essence,
but by the unity of his person.
For just as one human is both rational soul and flesh,
so too the one Christ is both God and human.