How do ovulation kits usually work?

Hormone detection. Ovulation kits detect a surge in lh, luteinizing hormone associated with ovulation. The more expensive kits also test for estrogen, which rises a few days before the LH surge, giving earlier signals of ovulation.
Detect LH surge. They detect urine lh, which is nature's trigger for ovulation once a follicle is mature. They are less reliable at predicting ovulation itself. It's assumed when lh is seen ovulation follows in the next few days. This is true in textbook cycles, but is variable in real life. Ironically, they work best for women with clockwork cycles, and much less well for women who are more irregular.