You will need to sign up for the service then tweet as you normally would. For tweets you'd like removed from your timeline after a set amount of time, you will have to include a hash tag such as #15m, #5h, or #7d. The hypothetical tweet would then be deleted after 15 minutes, 5 hours, or 7 days. You get the point.
Having tweets automatically removed from your timeline is a convenient way to clean up your feed, as well as remove outdated invites or information. A particularly useful example of when the service would be of great benefit is during an emergency situation. As someone who has lived in Colorado over the last two summers and dealing with large wildfires, getting information from Twitter has been vital. But often that information may be retweeted days later, leading to confusion. By using Spirit, emergency personnel can include a self-destruct time with a tweet, and not only will the original tweet disappear, so will the retweets.
To sign up for Spirit you'll need to visit its Web site and connect it to your Twitter account. Currently the developer is letting users in slowly, so you'll have to wait a little while for access. According to the developer's Twitter account, it's close to being able to open up the service to everyone.