Each caterpillar unravels a single strand of delicate raw silk from its salivary glands and wraps the minute thread thousands of times around its body in a figure of eight pattern.
In this context the word "minute" means very, very small. In this sentence, it is talking about how small/thin the strands of silk are. Synonyms of "minute" are "miniscule" or "tiny".
This can be confusing, because the word "minute" has a variety of other meanings in English, including as a unit of time ("there are 60 minutes in an hour"), as units in geographical coordinates/bearings ("the ship plotted a course towards 2 degrees, 10 minutes east") and as a form of written records of proceedings during meetings/conferences/etc ("the secretary kept detailed minutes of each of the board's meetings").
However, while they are spelled the same, these latter two senses of the word are pronounced differently in spoken English. When referring to something tiny as "minute", the word is pronounced like "my-noot" whereas the other two senses are both pronounced more like "min-nett".