Ātma is that which is present in all three states of experience, and without which these states of experience are not possible. Sāksi means witness which is unaffected by any states of experience. These 3 states are:
Jagradavasthā (waking state)
Waking state is a state where the involvement of the gross body, and subtle body consists of 5 organs of perception, 5 organs of action, 5 pranas, mind and intellect are there to respond from, and express to the world. Therefore ātma is said to be jīva in its complete manifestation here in waking state, where we can use our free will as a doer (kartr) to pursue our goals of life.
The svarūpa, nature of ātma is consciousness, and every experience implies consciousness. Without ātma there is no experience whatsoever. Without identification with the gross and subtle body, there is no waking state for ātma at all. But in the same time ātma is only a sāksi witness for this waking state. This fact can be explained when one goes to sleep, completely dropping of identification with sthūla and sūksma śarīra, one is still pretty much there aware.
Svapnāvasthā (dream state)
Dream is the world which is experienced by us during the time of sleep. Why don't it said that when you are not awake, you are dreaming? Because from the waking state, there is no direct connection to the dream. There is no sudden dropping of the waking state, and getting into the order of dream reality. From waking state which is involved identification with the physical body, need to be totally cut off which is called deep sleep state. And from sleep, we wake up half to dream. The dream-reality-experience is the same as in the waking state. When we dream, our expression there is complete. We are affected by the dream world and we affect the dream world. In dream we are undergo experience in which there is sūksma śarīra, but not identify with sthūla śarīra. We do not identify with the senses, but in the same time we are undergo sensory experiences, therefore ātma is a experiencer (bhoktr). Whatever experiences we go through during waking state are stored behind in our memory which is called impression (vāsanā). And from this impression, is what we are experienced in the dream. They may not appears in the same form, but in a symbolic form which is known to you, they are never unknown.
Susuptyavasthā (deep sleep state)
Deep sleep state is where ātma dropping his identification with sthūla and sūksma śarīra and only identifying with the kārana śarīra. Because of given up the individuality together with its problems (even a blind man is not blind when he is in deep sleep state), therefore everyone feel at ease (śantiśvarupa) after wakeup from deep sleep experience. Only causal body which is ignorance is still identified, therefore total ignorance is there while in deep sleep. Even though there is no other identification at all, but ātma the Consciousness is there as the witness, therefore one can say "I had a good sleep", after wakes up to this waking state.
From the analysis of 3 states of experiences, we can see that me ātma is presents in all 3 states but in the sametime not limited by them. I am not the waker, dreamer nor sleeper. I am the witness of these 3 states of experiences, without which these states of experience are not possible.