Essentially, what this label means is that everything you type in that search box goes to Google, Google does the searching, using its own index and ranking algorithm, and the results then come back to the website. So what you see ends up being the same thing you would see if you just searched on Google directly.
Google's search service can be configured with additional settings and filters by the website.
In most cases, if a website offers search powered by Google, what this means is that it has a default filter that makes Google only search for webpages from that website. This isn't some special functionality that Google gives websites. This is the same functionality that is offered by the site: search operator on Google, so you could just type site:example.com on Google if you wanted instead of using the search box powered by Google on the website example.com.
Some search engines like Kiddle are powered by Google. There are many child-safe search engines like it, which will, at minimum, enable safe search by default with no way of disabling it. However, in Kiddle's case, the service also provides some extra functionality: Kiddle only shows results from their own whitelist of websites for kids, and it won't search for unsafe words.
If search engine is powered by Google, that means it doesn't have its own index, and likely not even its own search algorithm. In other words, it isn't a real search engine at all. This means you can't find anything on it that Google doesn't already have, and the top results for a query are going to be the same results you would get from Google as well. The same thing would happen if a search engine is powered by Bing instead. See this list of search engines that have their own index for further reference.
The label "powered by Google" can also appear in websites that make use of data from other Google products besides web search. For example, Google offers Google Maps, so if there's a map widget from Google Maps, or some location data is used from Google Maps database, it may have the disclaimer that says it's powered by Google because the data and technology comes from Google.
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