The only army to really successfully transition into new military tactics was the Wehrmacht ( these tactics are pretty much the same as the current tactics deployed by advanced armies).

How did the transition from total trench warfare of WW1 develop into the military tactics of WW2?

Trench warfare was bad for every parcitipating nation because of the big amounts of casualties but it was worst for the nation with less ressources or time or both. Namely the Germans. Germany was naturally encircled by great powers and always tried to prepare to fight them all at once which was a necessity. WWI is a good example, fighting Russia, Great Britain and France was expected and plans were made to win fast in the west to take on Russia without a second front. This premisse let germans to focus on overcoming the trench warfare to be able to achieve fast decisive victories again ( Franco-Prussian war ). Obviously they also naturally wanted to improve tactics like always. The Germans were not the only military to think about what we later called "Blitzkrieg" ( combined arms or whatever you want to call it ) but British and French advocates had more trouble convincing their superiors.

What was the change which overcome trench warfare?

In short: Mobility, tanks and combined arms with surprise.

Trench warfare like WWI became obsolete because heavy breakthrougs were more likely to succeed because of the greater and more accurate firepower created for example by aircraft. A breakthrough with fast moving troops could just bypass enemy strong points and disrupt enemy supply lines and raid the back of the enemy troops. New trucks and various combat vehicle made it possible to exploit breakthroughs faster. On top of that the trench in itself became less usefull against tanks since they would just breach the line and the casualties normally inflicted by machine gunes didn't happen in these numbers. Paratroopers also attributed to the end of extensive trench warfare and made forts becoming somewhat obsolete, Eben Emal comes to mind.

Mobile combined armed attacks have to be met by mobile counterattacks ( attack on flanks encircling the spearhead ), this is pretty much the only thing which has been prooven usefull against such attacks ( Second Kharkov, Operation Saturn for example ) or just plain extreme superiority in numbers ( Kursk comes to mind ). These counter attacks naturally need big reservers which are kept behind and ready. Extensive trench warfare binds a lot of troops.

Also a natural reaction to the mobile and fast nature of the new attacks was giving land to be able to counterattack later when the enemy advanced too fast and stretched his lines to much. Good example here on smaller scale would be the southern counterattack of the red army at Kursk. The german counterattack while having not much ressources inflicted serious damage because the enemy advanced to far. But how do you want to give land if all your troops sit in trenches? You can't abandon a trench every week and build a new one thats not ressource efficient. The russians survived the german onslaught during 1941 partly because they could give land. Well not every country have this possibilities obviously.

World War II happened 20 years after World War obviously a lot of new technologies were developed and all of them factored in the new reality of warfare. As we all know the french didn't adapt well to the new reality. Without going into to much details, mistakes like using tanks primarily as infantry support made the french lose in 1940.