Typefaces become popular for a number of reasons, partly technology (which often drives fashion -- "Because I can" is a more potent driver than most people realize), partly the cultural milieu within which they fit and become associated, partly the mood they invoke (or don't). The grotesks in general arrived on the typographic scene at a time when Western culture was rejecting the ornate styles of the late 19th Century across all fields of design, including architecture and fashion. If you wanted to look modern, simplicity was the watchword.
Helvetica is tremendously precise, as befits a Swiss-made typeface, and is in many ways the ultimate expression of the grotesk style. Precision is terrific in the context of corporate communications, word marks and logotypes, much in the way that Greek columns and massive stonework are useful in the context of bank and government architecture. -- It sends a message. Precision conveys dependability, a conservative culture, authority, control; so the association with corporate identities, in hindsight, was probably inevitable.
This precision is much of what designers love about Helvetica, and also much of what they hate, because with precision comes inflexibility. Helvetica doesn't take kindly to tweaking.
An example from a different era might help put things in context. Neo-classical architecture and design gained tremendous popularity 18th Century for its symmetry and serene harmony. It was also part of a cultural rejection of over-ornate design (Rococo), founded in the rediscovery of the proportionality and harmony of ancient Greek buildings. It eventually fell out of favor, ironically, for the same reason: a Robert Adam dining room was so exactly proportioned and symmetrical that there was no way to modify it without "breaking" it. You couldn't change a chair, a rug, or the position of a table, without things looking uncomfortably out of place.
Another chapter in the Helvetica story came with the first Apple laser printer and Adobe Postscript. Helvetica and Times Roman were the very first licensed, digitized typefaces. Postscript hinting made them look good at the low 300 dpi resolution available at the time. They were "free," since they came with the hardware, which conventional type certainly was not. Designers took to DTP like ducks to water, and brought Helvetica with them.
In pre-DTP days, other large sans-serif type families such as Univers were just as popular as Helvetica for corporate work. Once DTP took off, however, Helvetica had a definite edge.
The size of the family -- the number of available weights and widths -- is also a factor in the choice of a typeface for a corporate identity, because it allows text to speak with many voices without breaking harmony. As a designer, I always try to err on the side of too large a family rather than too small when I'm choosing a typeface, if it's one that will be used in a large range of contexts over a long period of time. Myriad and Futura are also large families, and I use both, but the mega-families are even larger.
@bemdesign's answer is spot on. Whether Helvetica is good or bad for a particular project depends on the project. It is dry, passionless and, in most weights, severe. It's notable that Apple's corporate face is Myriad, for all that they liberally sprinkle Helvetica across their product line.