..How can we solve this question?
Suppose that y'=v, v'=F(y,v), where F,Fy and Fv are continuous for all y,v.
(a) As t increases, why do all orbits above the y-axis move to the right and orbits below the y-axis move to the left?
"Above the y-axis" means that v>0. Therefore y'>0, which means that orbits are moving in the direction of increasing y (in other words, "to the right").
(b) Why do nonconstant orbits that cross the y-axis always do so perpendicularly to the axis?
On the y-axis, v=0 and so y'=0. That means that the component of velocity in the y-direction is 0, so the orbit must move in the v-direction (that is, "perpendicularly to the [y-]axis").