Cloaking affiliate links is the technical term for making them look less ugly when they show up - usually in the bottom left of your browser - when someone hovers over the link.
Cloaking links is used by lots of sites and it's not just affiliate links that get cloaked. Google's search results hide their long and tortuous tracking data so that when you hover your mouse over any particular link, all you see is the link that's shown up green in the search results.
That means the average Google searcher probably doesn't know the extent to which they are being tracked and watched. Of course, this is being used to help improve the search results as the tracking data will feed back how long someone stayed on a site once they'd clicked the link which, in turn, implies how useful the result was and whether it justified its place in the results.
On your own sites, cloaking an affiliate link serves a couple of different purposes.
The first is the same as Google do: it's to make the links look less ugly.
Most affiliate programs give you a link that helps them keep track of who referred the visitor and, ultimately, who helped them make a sale.
That link varies considerably from program to program but is almost never as straightforward as something that looks like a regular URL.
There's often a '?' in the URL, followed by other information.
Amazon's affiliate links are incredibly long and full of all sorts of gobbledegook that only really makes sense to Amazon but allows it to make sure the click goes to the correct page out of the millions of possible pages on their site.
Cloaking that kind of link means that your site visitors are less likely to get distracted. Most probably don't consciously notice the long link that's shown in their browser - the major browsers like Chrome, Firefox and Internet Explorer have downplayed the link in recent versions - but we process a vast amount of information in our brains, so something will definitely get noticed.
But that reason is probably more to do with aesthetics than anything else.
The other reason that you should cloak your affiliate links is that they can change or even disappear over time.
As your website gets bigger, it gets harder to keep everything up to date.
Using a simple link cloaking program such as the Wordpress plugin WP Shorties (the one that I use on all my newer sites) means that if a link changes, you can update it in one place and that change is immediately reflected everywhere.
That's useful on a large site where it's easy to overlook one or two instances of a link.
It's also exceptionally useful if you're using your affiliate links elsewhere on the web - in a PDF ebook or on YouTube for example.
PDFs can't be changed once they've been downloaded, so if a link changes and you haven't cloaked it, that's it. You can't do anything about it.
YouTube video descriptions can be changed but it comes down to your record keeping and your time. The more places the link is, the more time it takes to change all the uncloaked links and the higher the chance that you'll miss one or more of them and miss out on the potential commission.