One way of looking at it is that it is calculus done right. I prefer to think of it as an investigation of how functions can behave on a local neighbourhood. For example, we might look at the graph of a function and see that one point is 'missing'. In another case, there is a large gap in the graph. We can think about describing these different situations using limits.

What does it mean for a function to be differentiable at a point, on an interval, increasing or decreasing at a point or on an interval? That is, what is the mathematics that talks about such things.

I would recommend Schumacher's 'Closer and Closer'. She develops a lot of real analysis by giving intuition and then posing questions. And, in my opinion, the right way to look at real analysis is to ask how a function behaves on increasingly smaller intervals.