With the brief information provided, it's not completely clear to me that another local business necessarily attempted to hijack your URL. There are numerous cases of Google associating URLs with businesses incorrectly.
For instance, I've seen cases where a domain which matches the business name perfectly, but is for a business in a completely different state, is associated in Google Places.
Make sure your business name, address and phone number are clearly displayed in regular HTML text on your homepage in the Title, Meta Description, H1 tag, and regular body text in order to reduce chances of errors occuring.
Just a couple of days ago, I saw where Google associated a website of a restaurant with their next door neighbor's business listing (a hotel), merely because the hotel's name was listed on their page ("We're next door to ____") and because of similarity in street address.
If this other business is showing up as claimed by the owner, then maybe this situation was caused on purpose. If your business listing/PlacePage is not claimed, that could also contribute to increased chances of error.
If the URL appears on both their business's Place Page and yours, it would be more likely to benefit you or not do much to you.