David is the writer of this Psalm. God is the Author of all of the Bible. David was directed to write this psalm as a reminder of the importance of having a right relationship with the Lord.
We know that David desired to build the Temple in his days. God told him he could not do so. He was excited about moving the Ark of the Covenant into Jerusalem. This was a great celebration for him and for the nation. Michal, his first and legitimate wife despised him for his behaviour at that time, but David was not trying to impress people with what he did. He was seeking to glorify God and rejoice in His blessings.
David was a busy man. He had a kingdom to govern. However, unlike his predecessor, he understood he could not govern Israel properly without following God’s law. He also understood he could not follow God’s law unless he was His child.
David wrote here that he was glad when they said unto him, let us go into the house of the LORD. David was glad for godly influences around him. He needed wise counsellors. He needed men who would keep him in touch with reality.
David knew that if he was grounded in God’s Word, he could stand as a king. He could face the challenges of being a king, successfully. He could know how to lead the people in battle. He could know how to lead the people in worship. Even though the Temple was not in existence in his day, the Tabernacle was there. There was a place known as the house of the LORD that he could go to.
It used to be that the presidents of the U.S. had a church that they were known to attend. We do not know if the current president even goes to church. For most of them it would not matter anyway. They did not believe in the true God. Our PM certainly would never encourage anyone to pursue knowing the true God. He would love to have everyone bowing before him but not the true God.
In verse 3 David wrote of the blessed state of Jerusalem. It was “compact together”. There was a unity within the walls of Jerusalem. David was able to build walls around the entire city. It was thus protected from enemy attacks. The people living in Jerusalem were united in purpose. Obviously that was not always the case. We know that Absalom tried to build a disunity within the city. He sat in the gate of the city and worked to turn the people away from king David and to himself. His rebellion was strong and devastating, but it was also short lived. God blessed David and He was there to heal the divisions that were brewing.
In verse 4 we see what compacted Jerusalem together. It was not David, it was God. When people know the true God and worship Him in truth, there will be unity. When the LORD is our king, there will be unity. Paul wrote of the importance of this unity in Philippians 2:

Philippians 2:3 Let nothing be done through strife or vainglory; but in lowliness of mind let each esteem other better than themselves.
4 Look not every man on his own things, but every man also on the things of others.
5 Let this mind be in you, which was also in Christ Jesus:

It is a good thing when people can be united in truth. It is a good thing when people do not need to act in an artificial manner of trying to make sure they do not offend someone with the truth. When the Lord Jesus Christ is in charge there will be unity.
We see much disunity among professing Christians today. We do not need to find a way to compromise the truth. We need humility to confess our sin and let God lead us in true unity. His desire is to bless His people with unity.

Psalm 133:1 Behold, how good and how pleasant it is for brethren to dwell together in unity!