This interpretation primarily comes from Revelation 12:9 and 20:2 without much regard to other passages, such as Genesis 3.

Revelation 12:9
So the great dragon was cast out, that serpent of old, called the Devil and Satan, who deceives the whole world; he was cast to the earth, and his angels were cast out with him.
 

Revelation 20:2
He laid hold of the dragon, that serpent of old, who is the Devil and Satan, and bound him for a thousand years;
 

These verses give excellent information about Satan and his many names as well as his involvement back in Eden, being the serpent of old. But does this eliminate that he used a real serpent? Not necessarily. The whole of Scripture needs to be consulted.

We read in Genesis 3 that there was a real serpent and it received a real physical curse to crawl on its belly and eat dust for the duration of its life (Genesis 3:14). Satan is not a physical being, although he can operate in the physical realm (Job 1–2). He is a spiritual being that operates in the spiritual realm as evidenced in many passages that detail his spiritual attributes, such as 1 Peter 5:8; Matthew 16:23; Acts 5:3; and Ephesians 6:12.

Luke 24:39
Behold My hands and My feet, that it is I Myself. Handle Me and see, for a spirit does not have flesh and bones as you see I have.
 

The Bible seems to portray Satan and his angels as disembodied spirits. So then, how can both Satan and a real serpent be the culprit? From other passages we find an important principle. Satan and demons can enter into people and animals and influence them. For example, Judas was entered by Satan in Luke 22:3; Peter was influenced by Satan¹ (Matthew 16:23); and the swine were entered by Legion, which consisted of many demons (Mark 5, Matthew 8).

Although such things may escape us, God easily sees when Satan is influencing someone and will often speak directly to Satan. Beginning in Ezekiel 28:11, for example, God is speaking to Satan who was influencing the King of Tyre. In the sections prior to this, the Word of the Lord was said to Tyre itself (Ezekiel 27:2), then to the ruler of Tyre (Ezekiel 28:2), and now a lament (expression of grief or mourning for past events) beginning in Ezekiel 28:11 to the King of Tyre. This one specifically was directed to the one influencing the King of Tyre—Satan—since the person, the King of Tyre, was never a model of perfection, nor was he on the mount of God, nor was he in the Garden of Eden, nor was he perfect in his ways from the day he was created, till iniquity was found in him (v. 15).

In Isaiah 14, the passage speaks to the King of Babylon and in some parts to Satan, who was influencing him. In Scripture, God sometimes speaks both to the person and to the one influencing that person—Satan.

So there is no stretch to understand that the Lord is speaking to the serpent and Satan in Genesis 3. Genesis 3:14 is said to the serpent and then Genesis 3:15 is said to Satan who is influencing the serpent.

Martin Luther states it this way:

Let us therefore, establish in the first place that the serpent is a real serpent, but one that has been entered and taken over by Satan, . . . ²

The Bible tells us that Satan used a real serpent to deceive Eve. And because of his entrance into the serpent, he can rightly be called the “serpent of old” or “great dragon” in Revelation.