You can contribute to a Challenge by clicking on the big green "Contribute" button on the Challenge view!

When you click on the "Contribute" button, a menu will appear and from there you can choose to contribute a new Record from scratch by selecting the Record Type from the list, or you can contribute an existing Record or Collection that lives within your profile by clicking on the 'View More Records' link! 

When contributing an existing Record or Collection, on the contribute screen you can search for the Record or Collection by name or, more directly, by typing in its "Record ID" (this is the unique number at the end of its URL, following the text "https://hitrecord.org/records/:" or "https://hitrecord.org/albums/".)