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I will make an educated guess, since it's a common mistake (and a user interface issue I don't know how to solve).

First click on the green New button. A new page opens where the necessary fields are in rows. Enter your redirection information there and click on the Save & Close button.

Now you're back to the main URL Redirection page. The fields you see above the list are filters. I suspect that you were previously trying to enter your redirection information in the filter fields which are used to limit the display of existing records, not create new ones.

If I guessed wrong feel free to record a short screencast going through the steps you followed to create a new record, e.g. using the free of charge Jing utility, and post the link to your video here. No need to speak a commentary, I would just need to see you go through the steps you normally take and lead to this issue.