The doctrine of God's aseity states that God is self-existent. He does not have it in him (whether inclination or power) to stop existing. He exists necessarily. Who made God? a child may ask. The answer is that he had no need of being made; he is always there. Or, for gown ups, God is a se (of or from himself). It is sometimes stated negatively: God is uncaused (the notion of self-causation should be avoided since it requires God to exist to actualize himself); and sometimes positively: he is, in himself, fullness of being. The doctrine is intimately related to divine simplicity (Day 1 above).
Again, the Divines are mimicking Patristic and medieval theology. Thus Aquinas: "But, in no wise does the supreme Nature exist through another, nor is it later or less than itself or anything else. Therefore, the supreme Nature could be created neither by itself, nor by another; nor could itself or any other be the matter whence it should be created; nor did it assist itself in any way; nor did anything assist it to be what it was not before." (Monologium, 6).
If pressed for Scriptural "proof", the Divines pointed to Jesus' statement: "the Father has life in himself" (John 5:26). As Calvin comments, "God is said to have life in himself, not only because he alone lives by his own inherent power, but because, containing in himself the fullness of life, he communicates life to all things." (Commentary, ad. loc.).