If there's a 75% chance of rain, would you say that rain is "likely"? Now reverse that: what percentage of rain possibility would it take for you to say that rain is "likely"? Maybe between 65% and 85%? Maybe at 86%+ you'd say rain is "probable"? Only at 100% could you say that rain is "certain."
But it's subjective. Different people, or domain experts (meteorologists in this case, perhaps), may use different terms like "likely," "unlikely," "a good chance," or "almost certain" for different ranges of probabilities. If it's not absolutely true (100%, or 1, or "on") or false (0%, or 0, or "off") then everything in between is a bit...fuzzy.
Computers, of course, deal with numbers and cannot parse "likely" for any real meaning. So fuzzy logic is a computer science way of collecting data from domain experts and plotting points in which they agree and then assigning points at which, say, 75% of them agree, or 25% of them agree. Connecting these dots forms line segments that can be measured mathematically. Knowing a value on the x axis, you can find the value on the y axis (and vice versa).
In AI systems, fuzzy logic is frequently used to evaluate confidence scores. In speech-to-text, for example, a sound is converted into a phonetic syllable based on the highest confidence score, and syllables are gathered into words based on the highest confidence score.