Why did Rome win the War?

Secondary historians

Modern historians explain the Roman victory in terms of a suit of 'reasons' - your textbook is typical:

Factors which enabled Rome to succeed:
 1.  superiority at sea;
 2.  roads and fortresses;
 3.  the loyalty of her allies;
 4.  the stability and determination of the Senate;
 5.  the co-operation of the people and their desire to win;
 6.  Fabius's strategy of attrition against Hannibal despite the destruction of the countryside;
 7.  the blocking of reinforcements for Hannibal;
 8.  the success in undermining Carthaginian power in Spain;
 9.  the superior discipline, numbers and organization of the army of Rome;
10. a military commander in Scipio who changed the way the army fought Hannibal.

Contemporary Writers

Historians at the time approached the question very differently to modern historians. 

Polybius was thoroughly rationalist - he was capable, for instance, of realising that, in their battle with Hannibal, the Romans had practical advantages: i.e. 'limitless supplies and inexhaustible manpower'.  But Polybius seems to have had a layered conception of causation which considered issues like this merely as the 'how' of why Rome won ... and that to explain WHY, you had to consider other more basic factors which underlay the Romans' victory - their constitution, their character, and their motivation. 
And that behind even that, of course, lay τύχη - the (?divine) driving force behind history - which had decided that the Romans would develop the qualities which led them to conquer the world. 

Polybius explains why the Romans won the war/conquered the world at the start of his History:

Polybius, Book 1, Chapters 1-3
No-one could be so unimaginative, so intellectually idle that he would not be fascinated to know how and under what sort of constitution in less than fifty-three years and all alone Rome came to conquer and rule almost the whole of the inhabited world...

My History will help readers to understand more clearly how they achieved it [and] the sources of their power and the motivation, which drove them...

[My readers] need to know what Rome’s ambitions were when she started upon this vast undertaking, where she got her resources from, and how she could afford it. It should be quite clear to my readers from that the Romans' calculations were thoroughly soundly based, both in their approach to the whole idea for world domination, and in their assessment of the resources required...

My own History has one unique quality derived from a remarkable characteristic of the period under consideration, and that is this: τύχη seems to have directed all the affairs of the inhabited world towards one single end and driven them to focus on one specific objective.

Livy's explanations were different.  I am not aware of any passage where he specifically addresses why the Romans won the war but - underlying everything he wrote - there are two factors which he seems to consider were paramount. 
•  The first was the Virtus Romana - old-fashioned Roman standards of behaviour and attitude. 
•  And the second was closely connected - an old-fashioned proper reverence for the old gods. 

In Livy the Romans fail when they forget these values, and succeed when they are led by people who value them.