Christ gives two clear distinctions about Himself in these verses. First, He says that He was not really a part of the Jewish nation, and further, not really of this world. Second, He says that if His Kingdom were of this world, His servants would fight for Him. The implication is that because they are not of this world, they do not go to war because His Kingdom is not presently established on earth.
This presents us with a vivid example of "us and them." The basis of this is that a Christian's loyalty is elsewhere. A true Christian sides with the spirit, while the merely professing Christian sides with the world, claiming to know God and to worship God, but denying Him by his works.
God has given us a clear command to come out of this world (II Corinthians 6:17; Revelation 18:4), and He clarifies it with examples such as this. In Philippians 3:20, He says through the apostle Paul that our citizenship is in heaven, and anyone familiar with the Bible ought to understand the legal ramifications of that. Not only that, we understand that His Kingdom is here only in spirit; it is not fully established as part of the earthly systems. The suggestion in verse 37 is that when it is fully established, if it were challenged, Christ's servants, true Christians, would go to war on its behalf, because our loyalties, our orientation, that which the true Christian sides with, are always with God's Kingdom. We will always rally to the things of the spirit.