The fields in the signature area are not digital signature fields, but rather are normal text fields, so it's intended for you to type your name. If the form did have digital signature fields you would be able to sign them. The type of security that they've applied does not permit you to create new digital signature fields, which is why Sign Document is greyed-out.
Download Cute PDF Writer, its a PDF converter. Convert the file you are working with to another PDF file by printing to Cute Writer. The new file will allow a printed & digital signatures.