They could legitimately be in need. It is odd that they can’t go to their friends/neighbors/family/local charity, etc. for help, tho. Myself, I’ve never been a big fan of e-begging, especially on craigslist. Technically the poor girls ad is miscategorized and should be flagged accordingly.
In any event, if you still feel moved to help, I wouldn’t send a personal check. She said she wants a full fridge, so send (anonymously) a gift card for a local grocery store.