For kids who have been introduced to exponents but haven't been taught specifically about what it means to take 'N to the zero power', this is an opportunity for a small 'Aha!' moment.
If a child already knows what is ten to the 2 (102 = 10 * 10 = 100) and 101 = 10, ask them what is 100.
Let them think a bit. Many kids will answer 'zero'. Ask, well, then what is zero times 10? If its not 101, then that can't be right.
Explain that 100 must be the thing that you multiply by 10 to get 101. This should be enough of a clue that they realize that 10 to the zero is 1.
Then, without any other explanation, ask something that sounds hard - "what is eighty-seven to the zeroth power?" With a bit of thought, your child may be able to come to the sudden realization that everything to the zeroth power is 1!
(This only works if they haven't already been taught this fact in school - it is fun to discover things that no one has told you. So don't be afraid to try this somewhat early, before most schools cover it.)