If you want the value of the {Contacts} field, omit the quotes around it.
IF(
{Business Name (from Contacts)},
{Business Name (from Contacts)},
{Contacts}
)
FYI, thanks for including the screen shot, but you cropped out the names of your fields in the screen shot, so it is a little hard to tell which field is which. I think that the first field is your formula field, but I cannot tell which fields are {Business Name (from Contacts)} or (Contacts}, although I suspect that they are not included in the screen shot at all. In general, it is best if you include all relevant fields in the screen shot, including the field names.