Why should I use DNS 8.8.8.8?

It is just another DNS server option.

Actually, it is the DNS server of Google, it means that Google provides the DNS and maintenance of this service, which means it is "more reliable" than some another DNS servers due to the fact that is maintained by one of the biggest IT companies of the world.

Also, according to Wikipedia, Google DNS provides some security functions that other DNS haven't:

The service does not use third party DNS management software such as BIND, instead relying on a custom-built implementation, with limited IPv6 support, conforming to the DNS standards set forth by the IETF. It also only partially supports the DNSSEC protocol.

Some popular DNS providers practice a form of DNS hijacking while processing queries,causing web browsers to redirect to an advertisement site run by the provider when a nonexistent domain name is entered, explicitly breaking the DNS specification. In contrast, Google's service correctly replies with an NXDOMAIN code in this situation, and as a result, many users are now using the service for this reason alone