Matthew 3:3 declares the divinity, the divine nature of our Lord Jesus Christ, that he is the Lord God Almighty. Whether it proves that Jesus is God depends on whether you believe that "all Scripture is given by inspiration of God" (2 Timothy 3:16). If you accept that then Matthew 3:3 does indeed prove that Jesus is God.
(Before going further I would urge you not to think we are all so many experts on Bible interpretation. In the Kingdom Halls everyone has the same understanding because they are not free to have their own. They are spoon fed and have to accept what they are taught: out here in the real world we are all encouraged to read the Bible for ourselves and reach our own verdict of the correct interpretation: this might sometimes need others to help us. Unfortunately there is out there in the Christian Trinitarian Bible-believing world methods of interpretation (hermeneutics) which have originated in unbelieving circles: I would recommend the book “Not Like Any Other Book” by Dr Peter Masters, cheap, easy to read, and a great antidote to the prevailing fashion in hermeneutics (which is dry, deadening, and essentially unbiblical in its approach).)
Also, before explaining why I think Matthew 3:3 proves Jesus is God I need to be sure you understand the various languages that the Old Testament and New Testament have gone through.
Hebrew (without vowels) OT becomes Greek OT (Septuagint Version) quoted in Greek NT becomes English NT
First it must be known that Isaiah was originally written in the Hebrew language. (The Dead Sea scroll version of Isaiah is in this [pre-exilic] Hebrew). This original Hebrew language was without vowel signs.
Then in the third and second centuries BC (300 BC to 100 BC), because the Hebrew language was not understood by many Jews, the OT Hebrew was translated into the Septuagint version which was in the Greek language.
The New Testament quotes of the Old Testament often quote from the Greek Septuagint version, including this verse in Matthew 3:3 and the equivalent in Mark 1:3.
Original vowel-less Hebrew changed to the Masoretic Text (vowel pointed Hebrew)
And then starting in about the 7th century AD and finishing in about the 10th century the Masoretic text (MT) was produced by Jews called the Masoretes. The Masoretes starting with the vowel-less Hebrew version added vowel signs to all the old Hebrew words.
The only exception was the name for God which we translate as Yahweh (or Jehovah). This written form of the name of God we call the “Tetragrammaton”. It might also be translated “I am”. The Masoretes, like many others, feared it irreverent to take the name of the Lord God upon their lips (and feared it too close to a possible breach of the commandment "Thou shalt not take the name of the LORD thy God in vain") so they did not add vowels to the Tetragrammaton, which they saw to be the true name of God. (We might choose the English letters "YHWH" or "JHVH" to represent the Tetragrammaton, from which we might pronounce it as either "Yahweh" or "Jehovah" (the anglicised form of "Yahweh", there being no "J" in semitic languages). Seeing as we do not have the vowels in the MT Tetragrammaton these would be little more than guesses of the original pronunciation.)
When the Septuagint was written the translators chose to translate the (original) Tetragrammaton into the Greek word "Kurios", which translates into the English word "Lord".
The consequence for Isaiah 40:3 & Matthew 3:3
The passage in Isaiah 40:3 down to 40:11 is clearly referring to the Lord God Almighty. As another has said:
”Nothing in the immediate context of Isaiah 40 suggests that Isaiah is referring to anyone other than Yahweh himself returning to Israel as king.”
(“Commentary on the New Testament Use of the Old Testament”, editors G.K. Beale and D.A. Carson, 2007)
The King James Version of Isaiah 40:3 says "Prepare ye the way of the LORD". And as has been said already "LORD" in the KJV is always a translation of the Tetragrammaton, the unpronounced name of God in the Masoretic Text.
In both Matthew 3:3 and in Mark 1:3 the Lord (Kurios) being referred to is our Lord Jesus Christ: he it is that John the Baptist prepared the way for. As many great kings throughout the centuries have had a herald to warn the masses on the streets of their approach, so John the Baptist was a herald for the coming of King Jesus, warning the people to be ready for his arrival. This, in itself, is useful to us: many a self-deluded man or self-seeking fraud has proclaimed himself to be the Messiah, but which of these has had a godly prophet to independently declare his coming beforehand?
In both Matthew and Mark the first part is a verbatim quote of the Septuagint of Isaiah 40:3. The NT quotes then end with changing “make straight in the desert a highway for our God” to “make straight his paths”.
In the NT verses “Prepare ye the way of the Lord, make his paths straight” the word “his” clearly refers to both Isaiah’s “our God” and also to Matthew and Mark’s antecedent “the Lord”.
Throughout the NT where Jesus of Nazareth is spoken of as our Lord Jesus Christ “Kurios” is in the Greek NT, the same word that is here used to translate the Hebrew Tetragrammaton in this passage.
Finally, the passage of Isaiah makes additional prophecies saying :-
“the glory of the LORD shall be revealed, and all flesh shall see it together.” How better than the glory of God being revealed in the person of our Lord Jesus Christ?
“Behold, the Lord GOD will come with strong hand … he shall feed his flock like a shepherd: he shall gather the lambs with his arm, and carry them in his bosom, and shall gently lead those that are with young” (Isaiah 40:11). This can be compared with Psalm 23 “The LORD is my shepherd”; with Ezekiel 34 eg verses 11,15 and with John 10:11 “I am the good shepherd” (which can even additionally be compared with Mark 10:18).
In Isaiah 40, then, and in other passages of the OT the LORD, Tetragrammaton, YHWH, God is the shepherd of his people. But in the NT our Lord, Kurios, Jesus Christ is the shepherd of his people who lays down his life in order to save his sheep. Jesus is the LORD. But elsewhere we read Jesus is sent by the LORD, that is his Father; and Jesus and the Father send the LORD, that is the Holy Spirit, for "the Lord is that Spirit, and where the Spirit of the Lord is, there is liberty." (2 Cor 3:17).
The Trinitarian God is love (1 John 4:6-21), one who will carry the lambs in his bosom, and gently lead those that are with young.
To be a Christian is to be a helpless sinner, in ourselves, assisted by the mighty power of God through his Holy Spirit, and by the reading of, and praying over, the word of God. By this means we shall be able to say "I can do all things through Christ who strengthens me" (Phil 4:13), though in ourselves always needing to be in the vine, for "apart from me you can do nothing" (John 15:5).