This number that Prism reports represents the value of the X variable when the Y value of the logistic curve is 0.5. Another way to say this is that this is the value of X where the model predicts the probability of observing a success to be 0.5 (or 50%). Mathematically, because:
P(Y=1) = 0.5,
P(Y=0) = 1 - P(Y=1) = 1 - 0.5 = 0.5
Odds = P(Y=1)/P(Y=0) = 0.5/0.5 = 1, and
Ln(Odds) = Ln(1) = 0
Using the formula for the simple logistic model,
Ln(Odds) = 0 = β0 + β1*X, we can solve for X:
β0 + β1*X = 0
β1*X = -β0
X = -β0/β1
Thus, for a given model, the value that Prism reports here is equal to -β0/β1. Note that this value has some additional interpretations in special cases. For example, if X represents the amount (or concentration) of some drug or other administered treatment, and Y=1 indicates the death of a subject, then X at P(Y=1) = 0.5 is the LD50 (or lethal dose, 50%). This value represents the amount/concentration of substance required to kill 50% of the test population. This is a very common value to calculate in toxicology.
In other dose-response type relationships, this value may also be referred to as the EC50 or IC50, depending on the specifics of the experiment.