The promise you quoted is true and it is also deeply personal. God is good, and because He is good He is a stronghold in the day of trouble. He does not promise that trouble will never come. Scripture says that as surely as sparks fly upward, man is born for trouble, and every life knows it in some form. It can be painful and even confusing when those who do wrong seem to prosper while you are trying to walk faithfully and find yourself pressed down.
What this verse declares is that when trouble does come, you have a place to run. For the child of God there is always a refuge. The name of the Lord is like a strong tower that the righteous can run into and be safe. He is not distant from your situation. He knows those who trust in Him, He knows the way of the righteous, and His ears are open to their cry. Simply to remember that God knows all about what you are facing can lift despair.
In the Old Testament God commanded cities of refuge to be placed throughout the land so that no one was ever more than a short distance from safety. At every crossroad there was a sign pointing toward refuge, so the way could be found quickly. Those cities were not for those who sinned with premeditation and malice, but for those who needed protection and a fair hearing, and as long as a person remained inside the city he was safe from the avenger. Scripture presents this as a picture of what Jesus Christ is for us. We have fled to Him for refuge, and in Him we are protected. One of those cities was called holy, pointing to Christ the Holy One, and another was in the heart of the land, reminding us that He is central and near. God has made sure that refuge is always within reach, and safety is found by remaining in Him.
That is why God invites you to call upon Him in the day of trouble. He has said to call and He will answer and deliver, so that you can give Him glory. You do not need a substitute when you have the living God to call upon. Much of our distress comes from our own folly and self willed ways, yet even then when we cry out to Him in truth He hears and His mercy endures. Because He is your refuge and strength, a very present help in trouble, you do not need to fear even when everything around you feels shaken, as if mountains were being moved into the sea. Your security is not in what can be shaken but in Him who does not change.
If you are feeling oppressed today, run to Him again and stay there. Trust Him, call upon Him honestly, and rest in the knowledge that He knows you and cares for you. He hears the cry of the oppressed and He will be your stronghold until the trouble passes.