This same Paul, who some few short years ago approved the murder of Stephen (Acts 8.1) and was still breathing threats and murder against the disciples of the Lord (9.1), now feels compassion for the Christians. He doesn't seem to have been a particularly compassionate person in his former life; he was zealous, a hard-ass, a no-nonsense Pharisee, high-ranking within the strictest of the Jewish religious groups. But his hardened heart of stone was demolished and replaced with a softer, kinder heart of flesh when he met Jesus Christ.
When he comes to know this Christ, then, Paul too is changed. He begins to embody the personality and interests of Jesus Christ. Just as he says to the Galatians that it is no longer I who live, but it is Christ who lives in me (Gal 2.20), so also here he says to the Philippians that he longs for them with the compassion of Jesus Christ. Paul is being transformed into the person of Christ, and begins to long for the Philippians the same way Christ does.
This is how we ought to be as well. We ought to be able to say that we long and care for one another with the compassion of Christ Jesus. It is only if we do this that we will be able to say, like Paul, Keep on doing the things that you have learned and received and heard and seen in me, and the God of peace will be with you (Phil 4.9). And if we are to have this kind of confidence and transformation, just as Paul, we need to seek to be filled with the Holy Spirit.