It's pretty basic, yet so easily dismissed.

Jesus said, "I will ask the Father, and He will give you another Advocate to be with you forever— the Spirit of truth... I will not leave you as orphans".

Later on in that very same discourse to His followers, Jesus explains in very clear language, the role of the Holy Spirit: 

However, when the Spirit of truth comes, He will guide you into all truth. For He will not speak on His own, but He will speak what He hears, and He will declare to you what is to come. He will glorify Me by taking from what is Mine and disclosing it to you. Everything that belongs to the Father is Mine. That is why I said that the Spirit will take from what is Mine and disclose it to you.

By itself, that explanation should probably be enough. He will guide you into all truth. Paul offers a similar description of how the Holy Spirit works in the life of a believer using slightly different wording. The Holy Spirit "speaks" to us by "testifying" or "bearing witness" with our own human spirit (Rom 8:16).