High rents is an effect of bigger corporate dynamics, not the cause.
Corporations generally locate the highest-priority projects in the head office to minimize communication costs. The executives are at the head offices, the largest concentration of staff is there, and you can pick the top employees from a larger pool. That means that if you're an ambitious young professional, you go to where the highest-priority projects are, because they will benefit your career the most in the future. This further reinforces the need to locate high-priority projects at headquarters.
I saw this play out at Google. The most ambitious, career-oriented young employees would move from NYC, Pittsburgh, or Boston to work in Mountain View, where they could have their pick of projects. Meanwhile, more family-oriented employees would move from Mountain View to Pittsburgh, Michigan, etc. to enjoy a higher standard of living and be closer to family. Who gets promoted with a big raise? The folks who work on the highest-priority projects. What does that do to area rents? It makes them rise in line with the highest-paid employees in that area.
You might ask why corporations don't move their whole headquarters to another location. Some do - Wells Fargo moved most of its operations from San Francisco to Minneapolis in search of cheaper labor. However, the same dynamic between employees within a company also plays out between companies within an industry. Google can't move its headquarters and operations because most of the interesting developments in the tech industry - startups to buy, startups to fund, new technologies to keep up with, key employees to hire - are in the Bay Area, and moving to say, Detroit means that they'll miss out on the cross-pollination of ideas that keeps them relevant in their industry. Note that this is not a serious issue for Wells Fargo, whose entire brand is based on "We're stolid and conservative, and we may not be in the forefront of financial innovation, but we will keep your money safe."
This dynamic played out a hundred years ago too - Detroit became Motown, USA because everyone involved in automobile manufacture was incentivized to move there, Hollywood became Hollywood because everyone in the film industry was incentivized to move there, etc. When those industries were vital and growing, they faced the same pressure on rents and economics as the Bay Area does today.