I'm a Tech Recruitment Manager here in the UK and I deal with oversees applicants all the time. I'll address your questions individually.
When I see the jobs that I would apply for (there are many) I just don't know how I am supposed to approach them.
Email the hiring manager directly with a short, personalised email. It needs to be immediately obvious that you're interested in a particular role and that you believe you meet the criteria but because of your nationality, you would require assistance with a visa. End the email with a direct question along the lines of "Is your company open to the idea of assisting me with a visa if I was deemed suitable for the role?"
Has anybody emigrated?
This one is probably best answered by a US citizen who has relocated to the UK.
Is it worthwhile applying to all the job ads?
Absolutely not. Most UK tech roles (more so in London) are inundated with applicants from all corners of the earth. You won't stand out or increase your chances. The same applies to recruitment companies. Most tech recruiters here in the UK are terrible and I would be amazed if any could appropriately assist you when it comes to visa issues.
How do you bring up the part about needing a visa?
It will be immediately obvious based on your nationality. As mentioned above, ask the hiring manager directly if they can assist. Most of the larger companies have an allocation of sponsorship visas however they are limited and therefore quite precious.
Is there a specific set of companies that are more open to this?
It's a bit of a minefield for UK based startups to hire US citizens but not impossible. Larger companies tend to have been through the process a number of times in the past and tend to have an allocation of sponsorship visas.
If none of the above what should I bury myself with learning to become desirable?
Ruby is in massive demand in the UK. Most companies with large ruby teams have no choice but to look abroad for talent as there are infinitely more ruby jobs in the UK than there are ruby developers. iOS, Python and niche technologies tend to be in big demand also.
My email is in my profile. Send me your resume and I'll do my best to give you some more tailored advice.