I do not understand what does it mean to say bundle of groups on a space? Does it mean that G as a set is disjoint union on groups indexed bybelements of X?

Presumably there's also a condition that the groups "vary continuously" in some sense, just like a vector bundle isn't any disjoint union of arbitrary vector spaces indexed by elements of a space.

The paper seems to give an example in example 1.3.

I do not understand the necessity of defining(considering) action of G on P  as map $G \times_X P \to P$

Well as you noted before G is a disjoint union of groups, so it's not going to be a group itself and therefore can't act on things. Instead, you want individual actions of $G_x$ on $P_x$ which "vary continuously" in some sense, and that's what a fiber product gives you.

I do not understand what does it mean to say bundle of groups on a space? Does it mean that G as a set is disjoint union on groups indexed bybelements of X?

Presumably there's also a condition that the groups "vary continuously" in some sense, just like a vector bundle isn't any disjoint union of arbitrary vector spaces indexed by elements of a space.

The paper seems to give an example in example 1.3.

I do not understand the necessity of defining(considering) action of G on P  as map $G \times_X P \to P$

Well as you noted before G is a disjoint union of groups, so it's not going to be a group itself and therefore can't act on things. Instead, you want individual actions of $G_x$ on $P_x$ which "vary continuously" in some sense, and that's what a fiber product gives you.

For general background on torsors, John Baez has a nice writeup in This Week's Finds which you can find with Google. (I'm writing on mobile.)