Alfred H. Barr was the first director of the Museum of Modern Art (MoMA) in New York City. MoMA opened in 1929. Around seven years later, Barr curated the exhibition noted in the question, Cubism and Abstract Art. The display encompassed around 400 pieces and took over all five floors of the MoMA building.
Cubism and Abstract Art showed that Cubism and abstraction were legitimate styles of art that couldn’t be dismissed as flash-in-the-pan moments that would shortly fade away. To further his argument that Cubism and abstraction held a crucial place within the evolution of art, Barr created diagrams that made connections between abstraction and Cubism and other forms of arts. These diagrams helped people see that abstraction and Cubism was a, more or less, natural outgrowth of art’s past.
The impact of Barr’s exhibition relates to global political events as well. As Barr was unveiling how certain modern artists were interpreting the twentieth century, forms of fascism were cementing their power in Europe, with Benito Mussolini in Italy and Adolf Hitler in Germany. These dictators often railed against modern art and enacted policies to repress the free expression that Barr’s exhibition evinced. It’s reasonable to say Barr’s show reinforced the idea that America was becoming a crucial place to present modern art as it was not systematically trying to censor and repress it.
Of course, regardless of its positive impact or influence, people still criticized the exhibition. They argued that it was too academic, obscure, and, despite Barr’s diagrams, that it betrayed art’s main duty to lucidly illuminate life.