‘He only is my rock and my deliverance,
He is my high tower, I will not be greatly moved.'
And this is why he can wait in silence before God in such confidence. It is because God is his Rock and his High Tower. Compare here Psalms 62:6, and see also Psalms 18:2.
God is his Rock. In other words He is firm and immovable, offering total security and a sure foundation. He is also his Deliverance. He knows that He will act on his behalf in order to deliver him from his enemies. Furthermore He is his High Tower, strong and unscaleable, the One in Whom he can feel absolutely safe. Knowing that his God has such attributes he knows that he will not be greatly moved.
Note the possibility that he will be moved to some extent. He is after all human. He may trip up but he will not be utterly cast down. We can compare Psalms 37:24, ‘though he fall he will not be utterly cast down, for YHWH upholds him with His hand'. So he is sure that with God on his side, such adverse movement will be unimportant and temporary. In Psalms 62:6, however, his faith has advanced and he is confident that he will not be moved at all.