How did Paul know that Satan hindered his movements?

Looking at the context, we see Paul, who often portrays himself as a proud Jew, talking about Jews in surprisingly harsh terms:

1 Thessalonians 2:15-16: Who both killed the Lord Jesus, and their own prophets, and have persecuted us; and they please not God, and are contrary to all men: Forbidding us to speak to the Gentiles that they might be saved, to fill up their sins alway: for the wrath is come upon them to the uttermost.

Raymond E. Brown, in An Introduction to the New Testament, page 463, queries whether this is an original part of 1 Thessalonians written by Paul or was it added by a later editor. Scholars have also sought to understand the wrath come upon the Jews "to the uttermost," and can see no historical issue that would have come to the attention of the Thessalonians before the First Roman-Jewish War and the destruction of the Temple in 70 CE, long after the death of Paul.

Burton L. Mack, says in Who Wrote the New Testament, page 151, he believes this addition to Paul's letter to the Thessalonians shows that some early Christians, angered by their exclusion from some synagogue, charged the Jews with killing Jesus and the prophets. If verse 2:18 is part of this interpolation, as seems likely, the reference to 'Satan' was really a pejorative reference to the Jews, implying that they had tried to prevent Paul from visiting Thessalonica.

I will attempt to respond using NT Scripture only and build my case accordingly, irrespective of any one denomination.

Context

The context is in the previous verses,

“For ...you suffered the same things from your own countrymen as they did from the Jews, who killed both the Lord Jesus and the prophets, and drove us out, and displease God and oppose all mankind by hindering us from speaking to the Gentiles that they might be saved...” ‭‭1 Thessalonians‬ ‭2:14-16‬

Hindering the spread of the gospel that might lead to salvation by the Jews

Connection

Who else hindered the gospel of the kingdom?

“"But woe to you, scribes and Pharisees, hypocrites! For you shut the kingdom of heaven in people's faces. For you neither enter yourselves nor allow those who would enter to go in.” ‭‭Matthew‬ ‭23:13‬ ‭ESV‬‬

Cause

And what does Jesus say about these Pharisees? Who is their father and who inspires their decisions?

“Jesus said to them, "If God were your Father, you would love me, for I came from God and I am here. I came not of my own accord, but he sent me. You are of your father the devil, and your will is to do your father's desires. He was a murderer from the beginning, and does not stand in the truth, because there is no truth in him. When he lies, he speaks out of his own character, for he is a liar and the father of lies.” ‭‭John‬ ‭8:42, 44‬ ‭ESV‬‬

Conclusion

So when Apostle Paul says

“because we wanted to come to you—I, Paul, again and again—but Satan hindered us.” ‭‭1 Thessalonians‬ ‭2:18‬ ‭ESV‬‬

If you put this all together earlier in verses 14-16 he says that the Jews prevented them from preaching the gospel to the gentiles and because they again have been hindered from preaching the gospel to them in person by the Jews, knowing that they are simply doing the will of their father the devil, ultimately it was the devil that was standing against them and not the Jews. Because the fight is not against the persons but against the power that guides the person

“For we do not wrestle against flesh and blood, but against the rulers, against the authorities, against the cosmic powers over this present darkness, against the spiritual forces of evil in the heavenly places.” ‭‭Ephesians‬ ‭6:12‬ ‭ESV‬‬

And in this case the Jews were being guided by their father the devil and Paul correctly identifies the source as being the devil. Fighting against the Jews would not have garnered as much success as fighting against the spiritual enemy dealing with the source directly.

As John MacArthur said in a recent radio broadcast, Satan and his minions are involved 24/7/365 in doing whatever is in their power to "kill, steal, and destroy" what God is doing in the world through His church and through the servants of His church.

Paul, being as he was on the cusp of doing great things for God by reaching out to the Gentiles with the redemptive message of the Gospel, was not immune from Satanic and demonic hindrances, and if Paul said "Satan hindered us," he was neither copping out nor engaging in the "blame game," but was rather stating a fact.

Satan and his demons and their ways are devious, evil, implacable, insidious, corrupt, immoral, crafty, wily, and much too powerful for even the most spiritual of believers to go "head to head" with them. Moreover, any attempt to "dabble" in the occult is perhaps almost as dangerous as attempting to engage occult forces in combat, which Paul was wise not to do.

The best we can do as believers is to put on the complete armor of God and simply stand against Satan in the power with which God supplies us (Ephesians 6:10 ff.). Prayer, too, can often be effective, particularly when combined with fasting (see Matthew 17:21 and Mark 9:29).

Perhaps God has gifted some of His servants with the gift and responsibility to cast out demons. Back to John MacArthur for a moment, he tells of a time when he came into the presence of a person who was demonized, and the demons communicated with John, telling him they knew who he was and that they wanted him to leave. From that one experience alone, John realized he was out-classed and out-gunned by the devil's demons, and he refused to engage them in battle. The wise man that he is, he, like Michael the archangel who disputed with the devil regarding the body of Moses, said simply, "The Lord rebuke you" and left the scene (see Jude, vv.9,10). We would be wise to do the same, unless God has called us to a ministry of what the Roman Catholic church calls exorcism. In the New Testament it is called simply "casting out demons" (see the list here).