Using bitcoins is an excellent way to stay anonymous while making your purchases, donations, and p2p payments, without losing money through inflated transaction fees. But Bitcoin transactions are never truly anonymous. Bitcoin activities are recorded and available publicly via the blockchain — a comprehensive database which keeps a record of bitcoin transactions. And when you finally use Bitcoin to pay for goods and services, you will of course need to provide your name and address to the seller for delivery purposes. It means that a third party can trace your transactions and find ID information. To avoid this, such mixing service provide the ability to exchange your bitcoins for different ones which cannot be associated with the original owner.
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.