Remember that exponents are just shorthand for, say, 2*2*2*2*2*2*2*2*2*2 (2^10).
Lets ignore the = 1 from the original eqn for the moment (the conclusion is still valid, you just have to follow the procedure above to get any constant).
y(t+1) = 2y(t)
Let's say that y(0) = c.
y(1) = 2 * c
y(2) = 4c = 2^2 * c
y(3) = 8c = 2^3 * c
From this, it should be clear to see that y(t) = 2^t * c, which is where the exponent comes from; as you're multiplying by 2 every time, after t cycles, you must have multiplied by t 2s, which is equal to 2^t.