As mentioned before, you could back up your tweets via TweetBackup, Twilert, TwapperKeeper or some other web apps.
But, because of the API limit, none of them were easy for you to back up all your tweets, especially when there were thousands of.
Now, Twitter is available for you to back up all your tweets through its website, and the following will show you how to do that in 3 steps:
1. Request your Twitter archive
After login, you can see a "Request your archive" button on the bottom of the Settings page, clicking on the button, you will receive an email when your Twitter archive is ready to be downloaded.
2. Download your Twitter archive
When receiving the email, you can click on the "Go now" button or the link below it to download your Twitter archive as a ZIP file.
3. View your Twitter archive
After download, unzip the ZIP file and open the index.html file inside, then you can view all your tweets by months.
For every backup tweet, you can view it on Twitter, reply to it, or favorite it.
A piece of cake, right? By the way, the downloaded archive will contain all your profile tweets, so use caution before sharing it if there are sensitive or secrete content.