It means the Spirit isn't just living in you; He's leading you, empowering you and overflowing out of you. Paul says it plainly: "Do not get drunk with wine, for that is debauchery, but be filled with the Spirit" (Ephesians 5:18 ESV). Instead of being controlled by a substance, be controlled by the Spirit.
When you're born again, the Holy Spirit comes to live in you (Romans 8:9). He convicts you, gives you new life, assures you that you're God's child, teaches you, and transforms your character with the fruit of the Spirit: love, joy, peace, patience and the rest (Galatians 5:22–23).
But the Bible also talks about being baptized in the Holy Spirit. John the Baptist said Jesus "will baptize you with the Holy Spirit and with fire" (Matthew 3:11 ESV). Before Jesus went back to heaven, He told His followers to wait for it: "You will receive power when the Holy Spirit has come upon you, and you will be my witnesses" (Acts 1:8 ESV). Then at Pentecost, "they were all filled with the Holy Spirit and began to speak in other tongues as the Spirit gave them utterance" (Acts 2:4 ESV). The same thing happened again in Acts 10 and Acts 19.
You might be thinking, "Do I have to speak in tongues?" Here's what I believe Scripture shows: not everyone will operate in the public gifts, like a message in tongues for the church. But it's God's will for every believer to be baptized in His Spirit and to have a prayer language. Paul said, "I want you all to speak in tongues" (1 Corinthians 14:5 ESV), and Jude says praying in the Spirit builds you up (Jude 20). God doesn't take over your mind or your mouth. It's a gift He gives, and you choose to use it.
"Isn't some of that stuff weird?" It can be when it's done wrong. That's why we need the Word and the Spirit together. The Word is the riverbank. The Spirit is the current. Remove the banks and you don't get more river; you get a swamp. Paul said, "Do not quench the Spirit... but test everything; hold fast what is good" (1 Thessalonians 5:19–21 ESV), and "all things should be done decently and in order" (1 Corinthians 14:40 ESV).
So how do you get filled? Ask. Jesus said, "How much more will the heavenly Father give the Holy Spirit to those who ask him!" (Luke 11:13 ESV). And keep asking. In Acts 4, the same disciples who were filled at Pentecost were filled again, and they "continued to speak the word of God with boldness" (Acts 4:31 ESV).
Scripture: Ephesians 5:18; Romans 8:9; Galatians 5:22–23; Matthew 3:11; Acts 1:8; 2:1–4; 4:31; 10:44–46; 19:6; 1 Corinthians 14:5, 40; Jude 20; 1 Thessalonians 5:19–21; Luke 11:13