Why did the office of vice president become more important with more responsibilities as the 20th century progressed

As the scope of responsibilities of the President of the United States increased during the 20th century, the Vice-President came to be relied upon to handle greater responsibilities also.

A 21st century Vice-President, Joe Biden, said it well:  “The way the world has changed, the breadth and the scope of the responsibility an American president has virtually requires a vice president to handle serious assignments, just because the president’s plate is so very full."  Biden was quoted in the book, The American Vice Presidency: From Irrelevance to Power, published by Smithsonian Books in 2014.  

Another point might be made about 20th century Vice-Presidents -- especially in the latter half of the century.  A number of Vice-Presidents became President because of things that happened to the President.  .  President Franklin Roosevelt died while in office, and  was succeeded by his Vice-President, Harry Truman (in 1945).  Vice-President Lyndon Johnson became President when John F. Kennedy was assassinated in 1963.  Gerald Ford rose from Vice-President to President when Richard Nixon resigned to avoid impeachment. 

In the current situation of American politics, with partisan politics strongly dividing Congress, Vice-President Mike Pence has functioned a number of times as the tie-breaking vote in the Senate on important matters -- another way the Vice-Presidency has taken on greater responsibility lately.

I think as the 20th century progressed, vice president has evolved from "the backup for president's mundane tasks" into the person who actually has enough power to make president's job easier
Unlike in the past, the vice presidents in this era usually  capable in handling heavy tasks that serve the House's agenda

hope this helps