FIV does not survive outside the cat's body and is only very rarely spread by casual cat to cat contact. The cat at greatest risk is an unneutered, outdoor male - he'll tend to roam, fight and mate, getting and giving bite wounds. The blood test detects antibodies against the virus. FIV is in the family of lentiviruses, and these viruses persist once a cat is infected - they don't go away. So the presence of FIV antibodies shows infection.
More false positives occur in low risk groups.
There can also be false negative results. A negative test can be repeated in 3-4 months to confirm it is a true negative. Kittens get antibodies to many diseases, in the mother's milk. Kittens younger than 6 months may have FIV antibodies gotten from the mother, without having FIV themselves. Of course, keep the cat strictly indoors to limit exposure to infection.