How does the anointing affect us?
There are only two or three passages that say we have been anointed.
“Now he which stablisheth us with you in Christ, and hath anointed us, is God; Who hath also sealed us, and given the earnest of the Spirit in our hearts.” (2 Corinthians 1:21-22)
“But the anointing which ye have received of him abideth in you, and ye need not that any man teach you: but as the same anointing teacheth you of all things, and is truth, and is no lie, and even as it hath taught you, ye shall abide in him.” (1 John 2:27)
“But ye are a chosen generation, a royal priesthood, an holy nation, a peculiar people; that ye should shew forth the praises of him who hath called you out of darkness into his marvellous light.” (1 Peter 2:9)
That last one does not mention anointing directly. However, there were three types of people who were regularly anointed: prophets, priests, and kings. Peter says we are a royal priesthood, so we have been anointed.
The first two passages above indicate that we are anointed with the Holy Spirit. Since the Spirit is the word of God (Ephesians 6:17), that anointing affects us in the way that the Holy Spirit works within us. That is, he helps us with prayer (Romans 8:26), he acts as proof to God that we have been bought (2 Corinthians 1:22), and he helps establish our faith (Romans 10:17). As we are filled with the word of God we are able to live more like he wants us to live, and we are able to praise God (Ephesians 5:18-21)