VERB+ing my way means performing the action of the verb through a series of locations or events.
For instance, a musician who had scheduled performances from the East coast of the United States to the West coast might say:
"I am playing my way across America."
A chef might plan a trip across France, with stops at restaurants operated by his friends, at each of which he planned to create a special meal. He might then say:
"I will be cooking my way across France."
In the instance you cite, the speaker is apparently an epicure (or at least a gourmand) who appreciates Italian cuisine. When he thinks about visiting Europe, he imagines enjoying meals in a series of Italian restaurants from one side of Italy† to the other, and sees himself as:
"Eating my way through Italy."
A harried teacher's assistant, staring up at the pile of unread essays left on his desk by a professor, might gripe to a colleague:
"Tonight, I'll be grading my way through these papers."
†A more prolonged and pleasurable gustatory itinerary would travel the country not across, but from North to South, beginning in Milano at Cracco and ending with a view of the sun setting over Messina, across the Ionian Sea, from the terrace Alle Cantine della Lampara in Reggio Calabria.