Prior to the advent of trustless alternatives, mixing services (also called tumblers) were used to mix one's funds with other people's money, intending to confuse the trail back to the funds' original source. In traditional financial systems, the equivalent would be moving funds through banks located in countries with strict bank-secrecy laws, such as the Cayman Islands, the Bahamas and Panama.
When mixing bitcoins, you send your money to an anonymous service and, if they are well-intentioned, they will send you someone else's tainted coins. So, now, whatever those coins were used for may now be traceable back to you. Additionally, mixing large amounts of money may be illegal, being in violation of anti-structuring laws.