How can humans lie to themselves?

It is possible to avoid recognizing a truth, which is already known to us on some level.

How is that possible?

The answer lays in the structure of our consciousness; we have a subconscious and a conscious mind. The subconscious contains everything we know and have stored away. The conscious is that content which we focus our mind on at any given moment. It draws both from the subconscious and from input from the world.

When there is something we experience in the world which is directly related to a low-level subconscious knowledge, we experience it automatically. We also experience emotions, which draw from the subconscious, automatically. However, to be conscious of more abstract or complex knowledge, we must focus our mind in order to perceive it.
People can spend years in psychotherapy digging into their subconscious to figure out what conclusions they have drawn and are drawing subconsciously. The content of our subconscious is not automatically conscious. If you've ever experienced an emotion which was based on a judgement which you could not decipher, Then you experienced first hand the separation between the conscious and the subconscious. Some content is buried so deep in the subconscious it takes a lot more than an act of will to bring to consciousness.

It is this fact that allows us to lie to ourselves. We can know something subconsciously but repress it and deny it - and thus make it absent from our immediate awareness.

A self-lie can only have so much power. Subconsciously, the knowledge is there. And that knowledge is being drawn and used to generate our emotions.
Moreover, when there is a collision between what a man is trying to tell himself and what his subconscious knows, he develops a sense of guilt and self-alienation.
Self-lies are possible - but they have their own nature - and they never go fully deep.

Imagine, for example, that a spouse cheats on their partner. They decide not to tell them about it and try to enjoy the love of the partner as if it were fully deserved and given by free choice and full knowledge. They lie to themselves when, during time with their partner they try to eliminate from their consciousness the nature of the choice their partner is making about them - an uninformed choice. They divert their mind from the fact that the affection they are receiving is not given by choice and that they do not deserve it.
When together, the spouse can divert their focus away from the betrayal and thus not experience negative emotions in an intense form. But subconsciously the knowledge cannot be erased, and therefore they cannot develop intimacy with their partner nor enjoy their love, nor feed on the partner's appreciation of them.

Animals cannot lie to themselves the same way that humans can't lie to themselves regarding very basic concepts. We cannot, for example, convince ourselves that a table we see is not really a table, but we can convince ourselves, to some degree, that we are motivated by a motivation which is different than the actual one. For example, telling oneself that one is motivated by a desire to help someone when in fact one wants something for oneself.

Animals cannot think in the way humans do - in that higher level, and therefore cannot lie.

So in conclusion, us humans can lie to ourselves by diverting our consciousness away from case-relevant and correct subconscious content, when that content is abstract and complex enough.
A self-lie can never go all the way because our subconscious generated emotions based on the subconscious content, whether or not we are aware of that content.
The emotion will always contain that which we hold subconsciously.