Performing at sight essentially means you're sight reading a piece. Imagine you're a pianist at a hotel or bar or restaurant. Someone comes over and hands you the sheet music for a piece you haven't practiced or seen before. You may or may not have heard the piece. Or, you're a church pianist, and the hymns for the service have been changed. Now, you have to play at sight rather than play a piece you've spent some time practicing.
Obviously this calls for a number of skills that I won't go into here, but a skillful use of the pedal can help you tide over any problems or fumbles you might encounter. When the music keeps sounding, few people notice your hands are off the keys for a brief moment while you figure out the next note and find it on the piano.
I once saw my teacher lift her hands off the keys to squint at a chord in a piece she was sight reading for me. Because she was using the pedal, the sound from the previous notes still sounded, and if I were just listening to her play, I wouldn't even have noticed.
Needless to say, you still need to master all the other skills necessary to sight read well. The pedal doesn't make up for a lack of those skills. And, remember, not all pieces lend themselves to the use of the pedal. But for those that do, using the pedal can often get you out of a tight spot.