Credible evidence is evidence that's likely to be believed. A credible plan is one that might actually work, and a credible excuse is one your parents might actually believe. And just as credible means "believable", the noun credibility means "believability". (But we no longer use incredible to mean the literal opposite of credible, just as we no longer use unbelievable as the literal opposite of believable.) Since cred is short for credibility, "street cred" is the kind of credibility among tough young people that you can only get by proving yourself on the mean streets of the inner city.