To start using this service, you'll need to connect your Facebook and Twitter accounts, and an in-game menu should allow you to do just that. In the event that this in-game pop-up doesn't appear, you can sign up manually under the game's "Settings" page at the bottom of the screen. You can then click on a check-box that says "FarmVille Twitter Enabled."
Once you're connected, you can send out as many Animal Feed and Watering Can requests as you normally do, and will see your requests automatically posted to your Twitter feed for other players to click on. Likewise, you can click on others' Tweets by searching for the hashtag (a categorization feature on Twitter) "#FarmVille." You can only claim five of these feeds within a 24 hour period, but none of this will have any effect on the way your Facebook news items work. Again, Twitter and FarmVille is a completely separate partnership from FarmVille and Facebook, so you only really need to connect your accounts if you're desperate for more Animal Feed and Watering Cans.
As one final note, if you have already connected your Twitter account to Facebook in general, this is not the same thing. You'll have to authorize a separate app (FarmVille@Zynga) on Twitter in order to make this work, so don't worry about auto-connecting without your approval.
Will you connect your Twitter account to your Facebook farms, or do you think the five-per-day limit makes the feature useless? Sound off in the comments!