The invisibility spray is kept in a rubber duckie until needed. When the liquid is sprayed on, it instantly renders the user (and their clothing) completely invisible. However, the effect is sadly temporary, and wears off after a few minutes. The spray can also be washed off, as Fozzie Bear learns when he washes his hands after using the bathroom during the rescue attempt.
Bunsen Honeydew explains how to use the invisibility spray.
The invisible Muppets leave footprints as they sneak into the compound.