In his soliloquy after Banquo leaves, what does Macbeth tell the audience he sees, and how does he explain the sight?

Macbeth's soliloquy after his conversation with Banquo in Act II, Scene 1 is intended to show Macbeth is subject to hallucinations. He thinks he sees a bloody dagger floating in the air in front of him, evidently leading him towards Duncan's chamber.

Thou marshall'st me the way that I was going,And such an instrument I was to use.

Macbeth finally decides the dagger is only an hallucination created in his mind by his emotional turmoil.

There's no such thing:It is the bloody business which informsThus to mine eyes. 

Shakespeare wanted his audience to retain some small amount of sympathy for Macbeth because the play was supposedly a tragedy and the audience should feel pity and understanding when the protagonist meets his fate. The "air-drawn dagger" is a way of suggesting Macbeth is really innocent, just a prey of all kinds of supernatural forces and his own mental problems.
Shakespeare tries in many ways to preserve sympathy for his tragic hero. For one thing, he makes it clear Macbeth doesn't really want to murder Duncan. Lady Macbeth plays a vitally important part in forcing Macbeth to commit the crime. She understands her husband. She says he is "too full of the milk of human kindness." She pinpoints his dilemma in her soliloquy in Act I, Scene 5, where she says,

Thou wouldst be great;Art not without ambition, but withoutThe illness should attend it. What thou wouldst highly,That wouldst thou holily; wouldst not play false,And yet wouldst wrongly win. 

In addition to ameliorating Macbeth's wickedness through his hallucinations and his domineering wife, Shakespeare also passes some of the blame for his misdeeds to the three Weird Sisters. There is also the fact that Macbeth bitterly regrets killing Duncan immediately after he has done so and continues to feel guilty until the very end of the play. He is plagued with guilt and afflicted with insomnia and nightmares. All this, too, is intended to create pity and sympathy for the tragic hero in spite of his murders and tyrannical rule. The playwright even has King Duncan himself suggest to the audience that Macbeth actually deserves to have his kingdom. 
Would thou hadst less deserved,That the proportion both of thanks and paymentMight have been mine! Only I have left to say,More is thy due than more than all can pay (Act I, Scene 4).