The term wasn't in the $.99 Webster's, so I went to dictionary.com, which said, "Existing independently of any cause."
So for God to be "self-existent" it means that God exists apart from any cause or thing other than Himself. God just is. God is because He is. There is no other cause for God other than that He exists, and the reason for the existence of everything else is from Him alone.
In Exodus 3:14, God said to Moses, "I AM WHO I AM. And he said, "Say this to the people of Israel, 'I AM has sent me to you.'" (ESV).
Moses wrote to God in Psalm 90:2,
"Before the mountains were brought forth,
or ever you had formed the earth and the world,
from everlasting to everlasting you are God."
Isaiah the prophet wrote the words of God when he said,
"I am the LORD, and there is no other,
besides me there is no God;
I equip you, though you do not know me,
that people may know, from the rising of the sun
and from the west, that there is none besides me;
I am the LORD, and there is no other.
I form light and create darkness,
I make well-being and create calamity,
I am the LORD, who does all these things."
"Turn to me and be saved,
all the ends of the earth!
For I am God, and there is no other."
And what was Jesus thinking when he uttered those unthinkable words?:
"Jesus said to them, "Truly, truly, I say to you, before Abraham was, I am." So they picked up stones to throw at him, but Jesus hid himself and went out of the temple."
Revelation 21:3 speaks of the new Heaven and Earth,
"And I heard a loud voice from the throne saying, "Behold, the dwelling place of God is with man. He will dwell with them, and they will be his people, and God himself will be with them as their God."
What a God... the only God! Can you wrap your mind around His self-existence?
Grace to you who have found this great God in Christ (2 Corinthians 4:6). I'm excited to know Him.