We need entities. To know that one exists requires knowing that there are existents. We need a link between future states, else man would not be equipped to survive. What we need not specify philosophically is the precise nature of the implementation of these requirements. Objectivism makes existence and consciousness primaries and declares both axiomatic.
Existence means the existence of entities that one can perceive.
This brings us to the issue of the relationship between existence and consciousness in the specific case of the relationship between a conscious mind and the body that houses it. Objectivism grants existence axiomatic status because it cannot be denied, is implicit in any claim, and can be gestured at and perceived directly. Present and future science has uncovered connections between mind and matter. These include the correspondence between mental states and brain electrical activity, the correspondence between damage to the brain and mental impairment, the explanation of the physical operation of the senses, the discovery and explanation of the operation of the peripheral nervous system and so on. But it is important to remember that the mental events are not their physical structure.