I think understanding that sentence reduces to parsing jargon. I'll provide an explanation I hope isn't in any way condescending. The main activity of research

I think understanding that sentence reduces to parsing jargon. I'll provide an explanation I hope isn't in any way condescending. The main activity of research mathematicians is theorem proving. That is, turning conjectures into theorems. In the jargon, proven theorems (and equations, bounds, counterexamples etc.) are often called 'results'. Essentially, the author is saying that with very few constructs and tools, number theorists have been able to pose a conjecture that is comparatively simple to state and understand, but is frustratingly as yet unproven, in spite of decades of effort by many, very good mathematicians. So far, many special cases ('partial results') have been proven, but the highly desired goal that is the proof of the BSD conjecture, in full generality, has not yet been achieved. The consensus is that mathematicians are quite far away from resolving the BSD conjecture, but very strongly believe it is true.