"At certain times the water in the Reversing Falls of St. John actually flows upstream or 'up the falls." What is meant by the above statement?

It means exactly what it says, but I'll try to explain it better. The Reversing Falls are a series of rapids on the St. John River flowing into the Bay of Fundy. (It's a drop, but it's not Niagara Falls.) The Bay of Fundy has a tidal range of about 52 feet, meaning that, at high tide, the water level has risen in the bay by about 52 feet. Well, at that level of high tide, the water in the bay is higher than the St. John River bed. Instead of water flowing from the St. John River into the bay, a so-called "tidal bore" causes water from the bay to flow back into the St. John River. Thus, the flow reverses, and the Reversing Falls live up to their name.

https://en.wikipedia.org/wiki/Reversing_Falls
https://en.wikipedia.org/wiki/Bay_of_Fundy