As I see it there are two ways (at least two) to interpret Paul's exhortation.
1. Disallow the Holy Spirit from doing miracles:
If this is what Paul meant, then "quench the Holy Spirit" means to "disallow" the Holy Spirit to act. His concern is that the Thessalonians might encounter supernatural actions of the Holy Spirit and discourage future supernatural encounters. When Paul exhorts the Thessalonians to not quench the Spirit, he indicates his intent as "do not despise prophetic utterances." If Paul meant "prevent" as in "do not prevent the Holy Spirit", then the phrase "do not despise prophetic utterances" would indicate same thing only in different words but only to add the motive as to why someone might disallow them. One might look down on those who manifest the Holy Spirit and treat them with contempt, because they have rejected the process altogether. This is how many today understand this passage, I think. Do not stop or prevent the Holy Spirit from acting out of your sense of contempt for such things.
2. ignore the message the Holy Spirit brings.
If this is what Paul meant, then quenching the Holy Spirit takes place when someone finds no value or worth in what the Holy Spirit has to say. His concern is that the Thessalonians might dismiss the message given to the apostles and prophets, which is why he follows his exhortation with the statement, "do not despise prophetic utterances." In this interpretation, the verb "despise" means "reject" on the basis that what was uttered has no value. He assumes that the prophetic utterances are being allowed to take place, but he also knows that people reject the word being given.
So, there is a meaningful distinction between our two interpretations. But don't get me wrong. I think both are wrong. That is, I think that it is unrighteous to disallow the practice of the gifts, and I also think it is wrong to ignore the message being given if it actually comes from the Holy Spirit. Both are wrong and both should be avoided. But the question on the table, I think, is which one did Paul have in mind when he penned 1The. 5:19-20?