I basically "just knew" when our current nanny walked in for her interview. She asked to wash her hands and then sat down at the table with me and was attentive while also making eye contact, smiling, and handing cheerios to my son in his high chair. They seemed to be comfortable around each other immediately, and she was the only nanny out of six that I interviewed that got on the floor with our baby.
That said, I don't think "the feeling" means that you will feel 100% confident and trust someone immediately upon first meeting - that comes with time. It just means that you got a good feeling about the possibility of working with the person and there were no major red flags.