"Z" could mean that he is eating the soup together with this guy. Basically, while English uses "with" when meaning "using", Polish doesn't use anything at all. "Z" with Instrumental will most likely be 'together with': On je obiad z żoną (He is eating soup with his wife"). Bare Instrumental is used in cases like here.
I would say that widelcem is the Instrumental case, and the rule I don't know, but its behaviour reminded me a bit of the word chłopiec, even the ending is similar. In case you're still learning this language, I would recommend to you to do it like I do, or find an explanation on the internet.
I gradually learn more good arguments in favour of all these additional grammatical cases we don't have in German. When I used to learn Czech in my university, I usually thought that the instrumental case solely existed to describe the usage of utilities, which didn't make sense to me, appeared to me like a scarcely useful case as is the Vocative. But in regards to this, it looks handily. Thanks for your clarification!
PS: Thanks, I use it as an “ersatz” to a profile picture like yours as I wore off of all the insults about my physiognomy. The internet can be a cesspit of negativity, I tell you. :D