In other words, before I listen further to all the preaching and jump into the manuals, what evidence do I get that this is the way to be successful in a successful project in a successful company?
There is empirical evidence that most IT projects are not successful (where success means on time, on budget and fully functional here). Given this evidence, it seems reasonable to wonder if a deterministic approach (the waterfall) is well suited for software projects.
"The definition of insanity is doing the same thing over and over and expecting different results." --Albert EinsteinRita Mae Brown1
If a deterministic process produces failures over and over, we are likely not applying the right process for software development projects and Agile methods are an alternative. The theory behind these methods is that most software projects are not deterministic, they are creative (like in art) and complex (as defined by Ralph Stacey) projects and we can't predict everything. So, instead of trying to predict everything and then fighting against change, we should use an adaptive process. And this is what Agile methods are about.
Now, using an Agile method will never guarantee systematic success (and someone claiming the inverse is a liar) but they'll give you better control over the risks. And, if your project has to fail, it will at least fail fast.
Update: 1 Actually, this quotation seems to be misattributed to Albert Einstein. The earliest known occurrence, and probable origin, cites to Rita Mae Brown.