At a higher level, think of decay rate the way that it is used in machine learning. The general idea is that the more recent the information is the more reliable it is.
In Sitecore, Decay Rate is computed using a half-life formula, where by 1 interaction is equal to 1 unit of time.
Real-world Example 1:
Consider a user browsing to the website of an athletic club that they are considering joining. After a few page views and interactions, this user is personalized and shown content geared towards a prospective member. The user is engaged and signs up for a membership.
Three weeks later, the user returns to the site. After a few page views and interactions, one might expect that this user would be personalized and shown content geared towards new members. This is where the Decay Rate comes in. If you are not using decay rates and you do not want to lose the data from your user's past interactions with the site, then from the moment your user returns your site he/she may still be shown prospective member content. If, however, you set up a decay rate, you don't lose any of that past interaction data and you still account for it in personalization. However the past data is worth less than the new data.
Real-world Example 2:
Another example is an e-commence site. Imagine you have a site that sells cellphones. If your visitor comes to the site and searches for phones and clicks on a bunch of the new ones then they are personalized and shown the newest phones. Assume that your user buys the newest Samsung Galaxy phone and leaves the site.
When your user returns, do you think your user is still interested in buying a new phone? Maybe, but the user searches for phone accessories. Now, what is the ideal experience to personalize the user with? Should they see iPhone accessories in our personalized advertisement components, or should we remember and account for the fact that on their last visit they looked for the latest tech, bought a new android phone, and chose the manufacturer Samsung? If we account for their previous visit's interactions, we can now show them phone accessories that best match the other data that we have from their last visit. As such, we may display the latest Samsung Galaxy phone compatible accessories first, followed by the latest in android and universal accessories, then older accessories and so on.
Summing it up:
The Decay Rate enables you to control the Rate at which past interactions decrease in value over time, using a basic half-life (1 interaction equals 1 unit of time) formula.
By effectively using Decay Rates, you empower your site to deliver more accurately personalized content, thereby improving engagement scores.