Using a mixer helps preserve your financial privacy. The question is: who are you protecting your privacy from?
Bitcoin transactions are public record, so anybody can view any historical transaction at their leisure. So when considering your privacy, there are a few different situations worth exploring, for example:
An unrelated third party is looking at your transactions
The blockchain history is public record, so anyone can view the entire history. Addresses aren't linked to real-world identities, but by looking at spending patterns, a third party can make guesses at which addresses are controlled by the same wallets (and thus users). Keep in mind this is only a guess, made by using certain heuristics that are not always correct, but nonetheless in many cases these guesses can still be accurate.
A counterparty to your transactions is looking
This is perhaps where a Bitcoin mixer becomes most useful. While addresses are pseudonymous, when you transact with someone they will gain some information about which addresses you control.
For example, if you have a 5 BTC UTXO, and spend it to send 1 BTC to somebody, then by looking at the transaction, your counterparty would know that you own at least 4 BTC more. And by analyzing the history of inputs (and outputs) from your transaction with them, the third party may be able to guess which other addresses you own, by using the same sort of heuristics mentioned above. So suddenly, just by transacting with someone, you've sacrificed a perhaps unnecessary amount of your financial privacy and history to them.
This is where a mixer can help: it breaks the links between addresses in your wallet, by allowing you to participate in transactions which break blockchain analysis techniques. As in the example above, your counterparty would still be able to see your change output, but they would be unable to look at the history of inputs (and outputs) from the transaction to gain more information about your wallet (though they would be able to tell you used a mixer).
An analytics company with a lot of resources is looking
This situation has the same considerations as above, but it is worth noting that analytics companies may have enough resources to engage in deeper investigations and analysis, for example by using IP address tracking to associate addresses to users. Nonetheless, using a mixing service will still help preserve your financial privacy.
As an example, many KYC'd exchange services are required by law to keep track of how their users interact with the service, and spend the bitcoin they acquire through it. This means the service might watch how you spend bitcoins that they sell to you, in order to ensure you aren't engaging in some illicit activities (according to the legislation the exchange must follow). So by using a mixer, you can sever the link between your exchange account, and your bitcoin spending habits.
Note that throughout all of this, the main idea is that you should only reveal your financial history to those individuals you choose to, meaning, this gives you the selective ability to preserve your privacy. If you need to provide a history of transactions for an audit, then you can do so, and a bitcoin mixer will not help you fake that history.
This makes a bitcoin mixer more useful for the everyday user that wants to uphold their privacy, than it is for a criminal wishing to hide their history from authorities. Maintaining privacy is difficult, but possible, and in fact many Bitcoin developers are working on technologies that will make the tracking of transactions more difficult. Maintaining privacy and fungibility is important: without it Bitcoin as a system of money and value would be broken.
EDIT: It is also worth mentioning that not all mixers are alike. In fact, many mixers have historically been unable to actually break the links between users/wallets/addresses, so they provide a false sense of security.
Consider this: if you were running a chain analysis company, and the existence of a mixer would hurt your business model, then why not just run a mixer yourself, and keep logs of all the transactions that users send through it? You would have front-row access to all their transactions, under the guise of providing them privacy.
With this in mind, A good mixer should be:
- Unable to distinguish between users/inputs/outputs, and thus unable to keep a history of users/transactions
- It should probably run exclusively over the Tor network
If a mixer doesn't satisfy these points, it should be avoided.
It is true that you can successfully hide your IP address and the origin of transactions using Tor; however, it is possible using the blockchain, the public ledger of Bitcoin to follow Bitcoin all the way back through transactions to where they were instanced being mined. This means that it is possible for someone to identify certain aspects of the movement of your Bitcoin, the equivalent of having someone watching where you draw your pay and go and spend it. Wallet linking is also still possible, and with one piece of information from where you have purchased goods or your Bitcoin exchange then a lot of information is available.
If we do not use a Bitcoin Mixer then more targeted attacks on the Bitcoin in our Wallet are possible through transaction tracing and the chances of them being stolen by more targeted attacks are higher.
A Bitcoin Mixer takes your Bitcoin and returns other Bitcoin in a non-linear fashion, breaking the linkage. This is useful. None in free countries should be concerned if governments know the movement of your Bitcoin, and it must be realised that there are researchers and people otherwise who also follow with interest. None would have them follow us around to watch how we spend our own money that we have earned.
Many Bitcoin Mixers may be found with [Great Big Internet Search Engine], however many may wonder if any are trustworthy. The following list is not exhaustive and I neglect to try them all, however I have tried using the Jambler.io partner program and found that it worked correctly when it was trialled: http://www.go-overt.com as you can verify for yourself.
You should still require to hide your IP address for privacy as discussed in answer to a separate question.