First, the subtle paradox is to be recognized in the words, "image of the invisible [i.e. not having a visible element] God." Immediately one considers that something else is meant by 'image.' A representation or display for sure, but not merely visual. An intimation of what God is, but not visibly.. but rather personally.
I'm reminded of a passage in Wisdom 7 which appears to have been in the author of Hebrews' mind in chapter 1:
Wisdom 7:24-27 (DRB)
For wisdom is more active than all active things: and reacheth everywhere by reason of her purity. 25 For she is a vapour of the power of God, and a certain pure emanation of the glory of the almighty God: and therefore no defiled thing cometh into her. 26 For she is the brightness* of eternal light, and the unspotted mirror of God's majesty, and the image of his goodness.
Hebrews 1:3 (DRB)
[The Son] Who being the brightness* of his glory, and the figure of his substance, and upholding all things by the word of his power, making purgation of sins, sitteth on the right hand of the majesty on high.
* The Greek word ἀπαύγασμα translated "brightness" (but which more accurately means 'effulgence') appears only here and in Wisdom in all of the Bible. Couple this with the unity of subject matter and we have a solid case for this being in the mind of the writer.
Wisdom is personified to a great extent in this Book (and indeed the other wisdom literature) in preparation for themes taken up and developed further in the New Testament (cf. 1 Cor 1:24). Notably, Wisdom is deified quite unhesitatingly, and without scruple: intended to show that Wisdom is spoken of as distinct from God more or less as a rhetorical device—God has never been without Wisdom (Jn 1:1; cf. Lk 7:35); and how could He have been?
God is invisible because an infinite and ineffable Being cannot have a literal shape or dimension by definition, and thus no 'appearance' that isn't percieved purely by other means than what we would consider 'vision.' The image of this God must therefore 'relate' or otherwise 'show forth' the nature of God, much like an icon relates doctrine by visible image yet doesn't pretend to comprehensively suffice as a representation of the figures (i.e. be the thing represented).
A very striking passage is found in John, where we read that the Apostles ask to be shown the Father. Note Jesus' response:
John 14:8 (DRB) Philip saith to him: Lord, shew us the Father, and it is enough for us. Jesus saith to him: Have I been so long a time with you; and have you not known me? Philip, he that seeth me seeth the Father also. How sayest thou, Shew us the Father?
Cf. John 14:7.
This is so interesting. Jesus is not the Father (Jn 15:26; Mt 3:17) but:
John 1:18 (DRB)
No man hath seen God at any time: the only begotten Son who is in the bosom of the Father, he hath declared him.
The word translated 'declared' is perhaps closer to 'explain' or 'reveal' or 'relate faithfully.' This passage explicitly precludes the notion that those that saw God in the Old Testament were seeing the Father, and that rather they saw the Son:
John 12:37-41 (DRB)
And whereas he had done so many miracles before them, they believed not in him: 38 That the saying of Isaias the prophet might be fulfilled, which he said:
Lord, who hath believed our hearing?
and to whom hath the arm of the Lord been revealed?
39 Therefore they could not believe, because Isaias said again:
40 He hath blinded their eyes, and hardened their heart,
that they should not see with their eyes, nor understand with their heart,
and be converted, and I should heal them.
41 These things said Isaias, when he saw his glory, and spoke of him.
Isaiah 6:1-10 LXX (Brenton)
And it came to pass in the year in which king Ozias died, that I saw the Lord sitting on a high and exalted throne, and the house was full of his glory. ... For the heart of this people has become gross, and their ears are dull of hearing, and their eyes have they closed; lest they should see with their eyes, and hear with their ears, and understand with their heart, and be converted, and I should heal them.
Cf. 1 Corinthians 2:8; Isaiah 44:6/Revelation 2:8.
The Son is God the Father to His creation. His Word. His manifest intent and communication otherwise unknowable (Mt 11:27). This is how we deal with the paradoxical 'image of [something invisible].' When God makes Himself known, that is the Son.