What does John 16:14 mean?

Christ is reassuring His closest disciples that when He leaves (John 12:32–33; 16:5), that physical absence will allow for the work of the Holy Spirit (John 16:7). Part of the Spirit's ministry is to "convict" the world so that non-believers will be confronted with the realities of sin, righteousness, and judgment (John 16:8). In the lives of believers, the primary work of the Holy Spirit is to guide and direct us towards the truth (John 16:13).

That work of the Spirit glorifies Christ, which means it glorifies God. At the start of the gospel of John, Jesus' ministry is described by saying "the Word became flesh and dwelt among us, and we have seen his glory, glory as of the only Son from the Father, full of grace and truth" (John 1:14). What we see in Christ is a representation of God the Father (Hebrews 1:3).

When the Holy Spirit teaches us truth, it is speaking of Christ (John 14:6), just as it is speaking of the Father (John 16:13). When Jesus refers to "what is mine," in this instance, it's a reference to those truths.