The gang obviously went to great lengths to conceal the Slayer's death, probably for several reasons. Primarily, if word got out that there was no one guarding the hellmouth, all hell would literally break out there. Also, probably of equal concern, if it were known that Buffy was dead, custody of Dawn would revert to Hank Summers if he could be located, or some foster home if he couldn't. Obviously, the Scoobies didn't want that to happen for Dawn's sake, but there was also something else: As far as was known, Dawn's identy as the Key might still be viable, which would therefore make her a target for whatever demon or wizard (Doc, for example) who wanted to utilize her power. That also explains why Spike was so adamant about protecting her.
Not much less important was the fact that the Watchers' Council, if they were to have learned about Buffy's death, would have almost certainly targeted Faith for assassination once more, in the hopes of attaining a more controllable Slayer after she was dead. Now, granted, neither Giles nor the Scoobies really counted Faith as their best friend, but they must have been aware that she had confessed to her crimes and was attempting to attone for her mistakes. They might not be certain of her reformation, but they were still unwilling to be at least partially responsible for her death at the hands of Council assassins.
Buffy was obviously buried in a secluded place, probably protected from discovery by some sort of cloaking spell provided by Willow and Tara. The Buffy Bot was repaired and put into service both as a substitute Slayer and as a substitute parent for Dawn.
Other than the half dozen people directly involved in Sunnydale and Angel and his team, no one ever knew that Buffy was dead. That's the only way that she could have resumed her life in Season 6.
