Deut/Dev 25:4 says,

Do not muzzle an ox when it is treading [grain].

Now Paul/Shaul quotes this verse in 1 Cor 9:9-10 and 1 Tim 5:18 in explaining why those who preach the gospel should be allowed to make their living from it. Why? Well, let's take an indepth look at this verse.

The Peshat

The Peshat of this verse is pretty simple. Don't muzzle an ox. You could discuss what "muzzle" means, but this is rather straightforward here.

The Remetz

Now we understand that one reason we don't want to muzzle an ox is because he might get hungry and need to eat and he might need that energy in order to tread. So he might need to eat some of the grain he is trying to tread.

There is a basic principle we can learn from this. That principle is not to get in the way of doing someone who is doing something productive. This alone would explain why Paul quotes this verse, since such a principle fits the context for which he used it in.

The Drash

The Hebrew word for "treading" here can also mean "to make fat". So we could also read this verse as saying...

Do not muzzle an ox when it is fattening.

The word "grain" is implied, but not actually in the text. "Fattening" would suggest we are not to muzzle it even when it is eating food it does not currently explicitely need.

Paul is drashing on this to say it applies to more than oxen, but also those who teach the gospel. Thus we'd conclude we are not to hinder those who preach the gospel by putting some obstacle in their way, even if the content of what they are doing does not seem necessary on the surface.

But wait - there's more to this drash. In the Aramaic, the word for "ox" here is "torah". Now it is not unusual for the translation of words from Hebrew to Aramaic to midrash on each other to some extent. Sometimes, when a word in one language means something different in the other, one is a symbol of the other. If we assume that's the case, we could read this verse as saying...

Do not muzzle "Torah" when it is fattening.

Now in Hebrew, "Torah" means "teachings". It does NOT literally mean "law", though we understand that many of these teachings are laws. Assuming Paul knew both Hebrew and Aramaic, we can see why he might have been so quick to have seen the spiritual meaning behind this law. Indeed, that may sometimes be why the Aramaic and Hebrew translate in this way - G-d may indeed have done this to make it easier to "decode" the Hebrew spiritual parallels.

There are other places where an "ox" is associated with "teaching" in Jewish tradition. For example, an "ox" is associated by tradition with the Hebrew letter Alef. The Hebrew word "Aluf"/"Iluf" - a cognant - means "to teach", "to train" or "to tame/domesticate". So here we have yet another place where an ox is associated with training/teaching. "Aluf"/"iluf" is usually used to describe training in an authoritative way - a master training a servant. An animal owner training his animal. The word "Aluf" can also mean "Chief" or someone in charge.

The Sod

Now getting to the Sod of this probably requires connecting ox with teaching. We see both connected to the concept of the "ultimate source" of something. In fact, the connection to "teaching" implies there is a master behind the teaching. Both concepts are connected to the Creative Force of Alef - the first letter in the Hebrew alphabet. So perhaps this is why, out of all the animals God / Elohim could have picked to use as an example, He picked the ox when He said...

The ox knows his owner, and the donkey his master's crib: but Israel does not know, my people do not consider. (Isaiah/Yeshayahu 1:3)

Now in some cases, the "master" may not be the owner. An animal can have an owner (who would be his ultimate "Master") and another "master" that trains him more directly, perhaps is paid by the owner to train him. So if we are conencting a "ox" in our minds with "Alef" or the "ultimate source" of something the way we should - and the way Hebrew thought tends to do - then we would certainly conclude that when we hinder someone like Paul from preaching the gospel, it is not Paul's work we are hindering but G-d's work. Because we would see Paul as only a trainer, hired by the Ultimate Aluf - G-d Himself - to train us.