In Native Son Bigger's principal fear, at least on the surface, appears to be that of losing face in front of others in the neighborhood. In the pool room, he provokes a fight, ostensibly because he believes the others are too cowardly to carry out the planned robbery of Blum's store, when Bigger himself doesn't want to go through with it. He then cuts the baize top of the pool table, causing the poolroom owner Doc to pull a gun and throw him out, because this is a presumably safer way of showing a kind of street bravery.
In my opinion, however, Bigger's greater underlying fear, and the one that drives all of his actions, is that he will never be able to escape from the neighborhood and the empty, violent life he leads. He dreams about flying a plane and revels in the detective stories he reads in magazines. He inwardly expresses a kind of admiration for the fascist dictators in Europe because they have the ability to whip their people up into a frenzy of action. Bigger has a despairing fear that the purposeless, poverty-stricken life of his family and others on the South Side is a permanent thing, that there is no remedy for it.
When Jan and Mary behave (without knowing they are doing so) in a condescending, insensitive way to him, Bigger's feelings of fear and anger are brought to a head. Though he does not intend to kill Mary, after he's done so he feels liberated in some sense, having created a new life for himself, though he knows that eventually he will be arrested and executed. Bigger's fears are thus confirmed; he is unable to escape the life he's been sentenced to by a racist society.