Hebrews 10:19 begins the verbal bridge that transitions from the doctrinal material to its practical application. This latter section contains arguably the most powerful exhortations in the entire Bible for us to get up and get going. If these Hebrews were not Laodicean as a whole, they were very close to it.
Overall, God is saying through the apostle, "Don't you realize your danger? Being justified and sanctified, you absolutely cannot allow yourselves to continue in your neglectful ways. You have powerful help available through Christ, yet you are drifting away! Don't you realize what you are giving up by your slow but steady drift into apostasy?" He had already warned them as chapter 2 opened that their neglect of their privileges and responsibilities was allowing this great salvation to slip away.
In Hebrews 10:19, He reminds them that they already have access to God, so they should come before Him with eager boldness. This is one of our great privileges. Adam and Eve were kicked out of the Garden and God's presence, but through Christ, God's regenerated children are now invited into His presence in spirit. Because the way has been prepared for us to do this, we are able to come to know God up close and personal. This is among the greatest of all blessings afforded to everyone who makes the New Covenant.
In other words, He meets with us, not outside the back or front door, but inside the house! And not merely inside the house but inside the second room beyond the veil—the Holy of Holies—where formerly only the High Priest was welcome once a year! The veil separating the rooms in the Temple was torn asunder at Christ's death (Matthew 27:51). Nothing hinders our liberty to go boldly into God's very throne room.
Jesus Christ Himself is "the Way" to the Father (John 14:6). As High Priest, Jesus has dedicated Himself to intercede on behalf of us sinners in our relations with God. In John 17:19, in His prayer the night before His crucifixion, He says, ". . . for their [His disciples'] sakes, I sanctify Myself." He set Himself apart to the shedding of His blood for us and to His position as our High Priest.
The phrase in Hebrews 10:20, "through the veil, that is, His flesh," refers to what He did as a human to make this access to God possible. When He was flesh and blood, He died for us so that we, like Him, could go directly into the Holy of Holies. Spiritually, His death pierced the veil.