API stands for Application Programming Interface, and it’s just that — an interface. Interfaces provide you with a sort of “hook” with which to retrieve and/or manipulate some underlying data. So when you say “view the entire API”, it doesn’t quite make sense in this case — that would just mean viewing the methods exposed by the API, which wouldn’t tell you anything about the possible data they can retrieve.
Some REST APIs do allow you to show all the data at once (often as a big ole JSON array), which may be helpful in some cases, but for large datasets it often wouldn’t help much. In this case, if the API allowed that, it’d give you a huge mass of data from as far back as they retain records and for all of the locations they track. You could spend many hours poring through it and still miss the rarely used “sandstorm” weather condition (just a hypothetical example).
Usually, a better bet is to read the API’s documentation… Which it looks like others have already linked to.