Isaiah 7:14
14 Therefore the Lord himself shall give you a sign; Behold, a virgin shall conceive, and bear a son, and shall call his name Immanuel.
This verse is quoted in reference to Jesus in Matthew 1:23, which says, "they shall call his name Emmanuel, which being interpreted is, God with us". Jesus is called "God with us" because He is God.
In the New Testament, Jesus is said to be the express image of God, and reveals God in such a way that He can say "he that has seen me has seen the Father" (John 14:9-11, compare John 1:18, 12:45: 2 Corinthians 4:4, Colossians 1:13-15, Hebrews 1:1-3), and it can be said of Him, "God was in Christ, reconciling the world unto himself" (2 Corinthians 5:19), and that "in him dwells all the fullness of deity bodily" (Colossians 2:9).
Therefore, Jesus is called "God with us" because He is a manifestation of the power of God, and a token of God being with His people. Consider the following verse:
Ruth 1:6
6 Then she arose with her daughters in law, that she might return from the country of Moab: for she had heard in the country of Moab how that the LORD had visited his people in giving them bread.
In the above passage, God being with the Israelites took the form of helping them meet their physical needs. It did not mean that God was actually walking around on earth. The New Testament Emmanuel is a similar concept. Someone could call their child "Emmanuel" today, and it would be appropriate, as it would be calling the baby a blessing from God, or naming them in honor or recognition of God's other blessings.
In fact, in the context of the original passage, the person in view is actually another "Emmanuel" who historically lived around the time of Isaiah:
Isaiah 7:14-16
14 Therefore the Lord himself shall give you a sign; Behold, a virgin shall conceive, and bear a son, and shall call his name Immanuel.
15 Butter and honey shall he eat, that he may know to refuse the evil, and choose the good.
16 For before the child shall know to refuse the evil, and choose the good, the land that thou abhorrest shall be forsaken of both her kings.
Above, the child named "God with us" was alive during the time of Isaiah, and signified a different instance of God being with the nation of Israel.
Therefore, Jesus is "God with us", not because He is God, but because God visited His people by working through Jesus, who He gave His Spirit without measure (John 3:34).