Looking at the three diagrams we can see that there are three possible situations

(a) mass is to the right of eqm
(b) mass is at eqm
(c) mass is to the left of eqm
(eqm = equilibrium)

Lets look at position (a)

If we consider the tension in both springs:
the tension in the spring on the left has increased by k_{1}x
the tension in the spring on the right has decreased by k_{2}x

So the overall restoring force SHOULD BE k_{1}x - k_{2}x
(the spring on the left is trying to pull it back to the left and the spring on the right is trying to pull it the right hence the forces act in opposite directions)

BUT according to my book the overall restoring force is k_{1}x + k_{2}x....how?

I don't understand the choices. There are two forces acting on the hanging mass: gravity and the force from the spring. Of course, a stretched spring exerts a tension. So I don't know the intention of having 'tension' vs 'spring' as separate choices: you could call that force the spring force or the tension force exerted by the spring.

And the given explanation reads like gibberish. Where is this question from?

Even if both springs are under tension all of the time, the restoring force would still be [itex] k_1x + k_2x [/itex]. There's an increased force to the left, because the left spring pulls harder and a decreased force to the right because the right sprint pulls less hard than in the equilibrium position. This will give the same effect as an increased force to the right.

A linear spring pulled from equilibrium exerts a force k*x. A linear spring compressed from equilibrium exerts a force k*x. Thus, the force is k*x+k*x.