^On the right track (groan), but not quite.
The benefit of goodwill is that it costs less money to buy rights to lay track and run trains in a territory. For small nations this cost may be inconsequential even if your goodwill is atrocious, but for very large and powerful nations, especially early in the game, it can save you hundreds of thousands of very important dollars to have good goodwill. An example of goodwill having considerable impact on the cost to buy rights is Russia in The Orient Express scenario - with Very Poor, Extremely Poor, or Atrocious goodwill, it costs about a million dollars to buy rights, whereas with Very Good goodwill you can pull it off for about $300,000.
Most of the time, there are no penalties for having low goodwill. Other than the cost to buy rights in a territory, goodwill usually has no effect, and once the rights are bought, you have them for good. The exception is some scenarios, which require certain goodwill as a winning condition, or has countries revoke your rights in their territory if the goodwill falls too low.
Goodwill is improved by delivering a large percentage of the loads available at your station. So if you have a bunch of stations in metropolises and aren't running any passenger trains, your goodwill will be low. Note that this element of goodwill is specific to countries - while it impacts your overall goodwill, it has a greater impact in each country. So if you're delivering great service in France but leaving all the passengers at the stations in Italy, the French will have higher goodwill than the Italians will with you. However, if you're delivering great service in France and Italy, Germany will see that and their goodwill with you will increase (albeit not to as high as levels as France's and Italy's) because they figure you'll give them great service, also.
Goodwill also can be improved by hiring certain managers who give bonuses to goodwill. The best ones for this are Ginery Twitchell (+15% goodwill, -15% stock price, $4000 salary), and Thomas Swann (+20% goodwill, $6000 salary). If you're planning to buy a bunch of territory rights, or need goodwill in a scenario, it can be well worth it to hire these guys. Twitchell gives you a nice opportunity to buy up your company's stock, as well.
Some scenarios have events that let you do something that'll improve your goodwill - usually at a monetary cost.
Finally, the worst thing that can happen to your goodwill in trains breaking down, or, worse, crashing. If you have several trains break down or crash in one year, you'll notice a significant decline in your goodwill. This is a reason both to replace your trains before they're ancient, and to consider trains with higher reliability ratings.
Hope that explains goodwill well enough. Railroad Tycoon II is one of my top 5 games ever in terms of time played, and though I haven't played it in several months, it's one of those I dig out and play again every so often just for the fun of it. Never did try any of the newer versions because RTII was so awesome.