Local chat is the chat that appears outside of group room chats and Personal Messages. The chat can be from avatars or from objects. The most common object chat is in the form of confirmational chat - such as confirming a purchase, or error chat - informing that an action taken failed to produce the results intended.
WHEN TO CHECK FOR LOCAL CHAT
When you take any action in SL and do not get the result you expected, that is a good time to check your local chat for error messages.
WHY IT MATTERS
These error messages will either tell you what to do to fix the issue, or allow a staff member to advise you on what to do to fix the issue. Knowing the error message is the quickest way to get you back on to pursuing your gameplay and will help lessen the stress on everybody. Sometimes even knowing that NO error message was received can help a Game Master diagnose the issue quickly for you.
Being prepared with the correct response when asked what the error message said is a good way to establish yourself as someone who pays attention and wishes to help themselves in the best way possible. These games are meant to be fun, using the tools SL provides to sort troublesome issues is a good way to keep them fun =)