I've been working in the tech industry for a couple decades now, so I've learned a bit about employee stock options. I'll stick a standard disclaimer here -- the tax laws surrounding them are moderately complex, and tax advice you read on the internet is worth what you paid for it. :) If you are in a position to have employee stock options then I STRONGLY recommend talking to a CPA to understand the implications.
The general idea with options is that the shareholders care about the value of the stock, and by compensating employees using company stock you are aligning the interests of the shareholders with the interests of the employees. Employees are more motivated to drive the stock price up, which is what the shareholders (who own the company and are thus the ultimate 'boss') want.
They are typically funded through dilution. That is, the company prints more stock and sells it to the employee when he exercises the option. No direct cash comes out of the company's bank account, but the larger number of shares results in a lower earnings per share, which will drive the stock price down by a tiny amount. The general idea is that this small decrease is more than offset by the increased employee motivation/productivity. It also results in a lower direct compensation expenses for the employer, leaving more money available for other things, and it helps in recruiting better talent. Sometimes companies will do stock buybacks where they use cash from profits to buy shares on the open market and retire them. If a company is doing this and also issuing new options, then one could look at the buyback as paying for the options.
Stock options showed up as executive compensation partly in reaction to public outcry about executives who got huge salaries even if they ran the company into the ground. They also get used in startup companies (usually tech companies) to compensate early employees (lower level employees, not just execs) who are often getting a much lower base salary than they would at a larger, more established company. If the company goes IPO, those employees can reap large rewards from the pre-IPO options, but fewer than 5% of tech startups ever go public so it's a gamble. In the 90s, even large, established tech companies that were already publicly traded (Microsoft, Cisco, Apple, etc) were giving stock options to most of their employees (again, not just execs), albeit in substantially smaller quantities than startups would do. That got largely shut down by a revision to the accounting rules, however. (Nowadays the bigger companies mostly use RSUs instead of options -- they're similar, but grant actual shares rather than options to buy.)
Taxation on stock options varies. There's two main types -- incentive stock options (ISOs) and non-qualified stock options (NQSOs). NQSOs are fairly straightforward, in essence the employee chooses when to exercise them (within the vesting rules) and from a tax perspective it's basically the same as if he got a cash bonus on that day. The employee does technically get the shares, but most people will elect to do a "same day sale" where the broker sells them immediately and just deposits the cash into the brokerage account instead. ISOs are more complicated, and have a tax treatment designed to incentivize exercising early and then holding the shares to get long term capital gains instead of ordinary income, but doing so risks AMT. ISOs can be quite dangerous in this regard -- in 2001 a number of people failed to do this and wound up owing the IRS more in tax than their (now crashed in value) stock was worth, in some cases by hundreds of thousands of dollars!
The "value" of an option is somewhat ill-defined. There's a formula to calculate it, but it's intended for use with market-traded options, not for employee stock options. Employee options are different and have a lot more restrictions, so the formula is going to give a value that's higher than it should be.
Option treatment when a company is sold is pretty variable, yes, sometimes they all vest immediately, but other times they get cancelled and replaced with options in the purchasing company. In some extreme cases they vanish and become worthless, it all depends on the fine print of the options agreement, and there was one case recently in which even shares from options that had already been exercised became worthless (google for articles about the Skype acquisition and how the employees got screwed there if you want to see what can happen). If you're in a position to get options in a pre-IPO startup, then you REALLY REALLY want to run that options agreement past a CPA to understand what it means.