Jim has hit the nail on the head; Live Local is a movement designed to encourage folks to support their local economies and businesses.
I'd only like to add that the Live Local movement is largely in response to expansion and proliferation of franchises across North America. Sometimes I've marveled how, just before I get on the highway, the last things I see in my hometown are a Best Buy, a McDonald's, a Lowe's, an Olive Garden, and a Target. I drive 300 miles and across two state lines, and the first thing I see as I get off the highway are a Best Buy, a McDonald's, a Lowe's, an Olive Garden, and a Target. There's even a term in the Urban Dictionary to describe this phenomenon – generica (which is a neologism of generic America):
What most of America, particularly Middle America, has become characterized by the ubiquitous Wal-Mart, TGI Fridays, Applebees, Chili's, Gap, J Crew, Best Buy, Target, Olive Garden, PetCo, McDonalds, Banana Republic, Border's, Costco, Home Depot, Nordstrom, Circuit City, KFC, Sports Authority, Starbucks.. and on and on it goes.... until every city/town/suburb starts to blend in and look like one big flavorless mall... where all the people, genericans, mindless zombies, consume and consume and consume.... slaves to corporations and keeping up with the Jones'...
As we travel through and across the country, it's becoming harder and harder to find mom-and-pop businesses among the restaurants and shops of a particular region. Live Local, then, is an exhortation to actively support such businesses, lest the last vestiges of local color die out from a community's restaurants, inns, and shops. It almost requires a certain grass-roots mindset to keep these businesses viable, because it's difficult for them to compete against the larger, more prominent chain outfits, which are usually much easier to locate and recognize.
Can I use it (live local) in my conversations with a native to encourage her/him in protecting local life in her/his hometown?
If "protecting local life" means "keeping local businesses healthy", then you could use that expression. However, it's not a particularly common idiom, and there are other ways you could express the same sentiment that might be more understandable. Perhaps the most common way to convey that thought is the phrase "Buy Local", but there are many catchy ways one could encourage the support of hometown businesses. I found these graphics by doing a Google image search on "support local businesses":
The one in the bottom right corner is a particularly clever pun. The expression "mind your own business" is usually an admonition to not meddle in the affairs of others. However, in this context, the word mind is being used in the sense of take care of (that is, the meaning of the word shifts from Definition #20 to Definition #22, as listed in this dictionary entry). While the pun is clever as part of an ad campaign, I would not use it in day-to-day conversation, lest it be misunderstood.