There are a lot of reasons. I test a lot of website apps, and I've learned that I get a lot of unwanted email from signing up for user accounts.
If I use a disposable email address, I can set up a user account, test the app, and see if I get tons of spam as a result.
If the website proves useful, I can cancel the "test" user account and set up one with my normal email address.
MakeUseOf.com Offers Options
I'm not going to reinvent the wheel. MakeUseOf.com, a website to which I subscribe, wrote an excellent article a few months ago about the various websites that enable you to create and use a disposable email address.
In Need a Disposable Email Address, they give 6 websites that offer that service and explain what each is best used for.
Do not mistake this for being anonymous online. Anytime you access a website, your IP address is automatically sent to the website (data packet). Read another article published on the same website, Can You Really Be Anonymous Online?.
Free apps are wonderful--especially when they fulfill a specific purpose.