If you the objective allows all calculations to be performed in Mathematica (rather than reproduced in Excel), then using the Interpolation function works great:

data = {{0., 0.}, {0.05, 0.4651}, {0.1, 0.6433}, {0.15, 0.7298}, {0.2,
     0.779}, {0.25, 0.81}, {0.3, 0.831}, {0.35, 0.8462}, {0.4, 
    0.8576}, {0.45, 0.8664}, {0.5, 0.8736}, {0.55, 0.8796}, {0.6, 
    0.8848}, {0.65, 0.8895}, {0.7, 0.8941}, {0.75, 0.8989}, {0.8, 
    0.9045}, {0.85, 0.9118}, {0.9, 0.9231}, {0.95, 0.9445}, {1., 1.}};
f = Interpolation[data]
Show[{ListPlot[data], Plot[f[x], {x, 0, 1}, PlotRange -> All]}]
f[0.756]

with output

But if you need the fitted equation outside of Mathematica (like in Excel), then determining how Mathematica deals with the interpolation object might be possible but I don't know if that's available. If providing predictions to others is the objective, does turning this into a CDF document achieve that?

Excellent about the use of StringReplace! I'm embarrassed to say that all these years I've been editing those output equations by hand. Thank you!

I wouldn't necessarily recommend polynomials at all. I should have asked initially how your data was generated. As a statistician, I'm always suspicious of data with such a smooth connected set of points. Is this one run of some experiment? Are the points generated by a much more complex formula and you're trying to approximate that with something that faster to run? Is the process that generates the data inducing a serial correlation? In other words, what is the data generation process? (I ask because LinearModelFit as used assumes that the errors are independent and identically distributed - which is important if you want to make inferences about the parameters or predictions.)