If you want a more in depth answer, in a word, if you hold that machine-made matzah is kosher for the night of Passover, then machine-spun tzitzit are probably fine, because in some respects, the issues are quite similar. In both cases, you are required to fulfill the mitzvah (of tzitzit on a four-cornered garment or eating matzah on the first night of Passover) with an object (tzitzit/matzah) specifically made with the intent to perform a mitzvah. This means if you happened to find just the right strings at your local hardware store, or discovered that a certain brand of crackers is kosher and made in under 18 minutes from start to finish, you wouldn’t be able to use them to fulfill the respective mitzvah, because they lack mitzvah intent.

The question is, if tzitzit or matzah is made with industrial machinery, is it considered to have mitzvah intent because the person operating the machinery has mitzah intent when he presses the start button? To understand the halachic issue, please refer to this blog post.

I never use machine-spun tzitzit myself, but when my boys are in a bind (they go through tzitzit very quickly) I often give them a tallit katan with machine-spun tzitzit until I or they have a chance to tie on hand-spun. My impression is that not a lot of people insist on hand-spun for boys under bar mitzvah age.