help with qualitative analysis of ODEs

Please can someone explain the following ideas to me?

Definition: the stationary point of an autonomous equation dy/dt = f(y) is defined by: F (y0) = 0 (so the points y such that the function is zero?)

y0 is a therefore a solution of the problem:

dy/dt = f(y), y(0) = y0

dont understand what this is saying?

A simple observation is that if f(x)>0 then dx/dt>0

and if f(x)<0 then dx/dt<0

why???

Definition: A stationary point is said to be asymptotically stable if for all dy/dt sufficiently close to y0, the solution of the problem dy/dt=f(y), y(0)=dy/dt

satisfies lim y(t) = y0 (as t tends to infinity)

Really have no idea what this is saying? And what is the purpose of qualitative analysis?