The National Security Agency (formerly known as Never Say Anything or No Such Agency) came into being from a variety of other preexisting organizations doing similar work in the Department of Defense. They were combined and given new resources because of at least four events (and possibly more) that made it abundantly clear to the Truman administration that this was important work. Three of these happened before the NSA was created, and the fourth was in progress when the NSA was founded.
The best discussion / introduction to Venona that I have found is the following preface on the CIA website. It is odd that it is on the CIA website but that is a nuance to be discussed only if you believe other parts of my post.