Adam Clarke on Matthew 10:20

For it is - the Spirit of your Father, etc. - This was an extraordinary promise, and was literally fulfilled to those first preachers of the Gospel; and to them it was essentially necessary, because the New Testament dispensation was to be fully opened by their extraordinary inspiration. In a certain measure, it may be truly said, that the Holy Spirit animates the true disciples of Christ, and enables them to speak. The Head speaks in his members, by his Spirit; and it is the province of the Spirit of God to speak for God. Neither surprise, defect of talents, nor even ignorance itself, could hurt the cause of God, in the primitive times, when the hearts and minds of those Divine men were influenced by the Holy Spirit.

Your Father - This is added to excite and increase their confidence in God.