The short answer is it all has to do with what tools were used to create the site. Every site you see may well look and work differently in different browsers. Your best bet is to have at least Safari, Firefox, and IE on hand. IE on the Mac is old - it hasn't been updated since 2003. I don't think you can even get it from MS anymore. Many users find that if Safari doesn't work well, that Firefox does. If you want to understand a bit more - keep reading. If not, download Firefox and give it a try, if, Safari seems to be not showing the site correctly.
Sometimes sites will post that text about only working with IE to cover their butts because they don't want to invest the time in testing and making sure their site works well from other browsers/environments.
If you understand that all internet content you see is an interpretation (of html, javascript, java applets, flash, CGI, PHP, etc.), then you also understand that the tools one uses to develop a web page/site have an impact on the code one creates for their page/site. That also impacts how well the site is interpreted. Basically what the site is saying is that they have only designed and perhaps tested for that particular browser's interpretation of their pages.
From a marketing perspective, if a site is smart, they will have research for their "typical user" - that is, what browser they are using (Safari, IE, Firefox, etc.) - from what environment (Mac, Windows, Linux) and what screen resolution (800 x 600, 1024 x 768, etc.). Sites that want to make it as easy for users to see the content as the creator of the site intends, will take the time to test the site considering these variables to make sure the end user has the best opportunity to see what the company wants them to see.
Lastly - in the case of a site giving out and letting you get info over a secure connection, the server software solution becomes a larger factor in the mix. Many of the server solutions targeted for small to medium sized businesses are Windows centric and sometimes use code that is less industry standard than the code used by other solutions. It's in Microsoft's best interest to make sure that at least their own browser works predictably with their server solutions, so they make sure that in general, it does. They have the benefit of knowing the idiosyncrasies of their own (server) software - other developers don't get the benefit of that, but instead aim for being more industry standard compliant. So, there are instances where the details of the server one has to connect to also have a direct bearing on what browsers will work.
Personally, I do a wide range of purchasing and managing financial stuff on line - with businesses large and small, including my credit union, which is a very small operation. Yet, with all of these, I have rarely had a problem using either Safari or Firefox to accomplish the tasks I needed to. I say this to underline that it really depends on the site and how customer friendly they want their site to be. That my credit union has worked so well, when I read of much larger companies creating sites that won't work well only underlines that it depends on the server solutions used and how industry standard they are. It also means that the browser developers need to keep their browsers up to snuff as the standards get tuned and tweaked to allow for new options and technologies. At the end of the day, the internet will always be about interpretation, so browser and site compatibility will always be a factor in how well a site works for a given user.
