Loess Fit and Resulting Equation
Loess doesn't give you an equation [1]. If you just want to get the values returned by the loess function you use predict(loess.object, new.data)
[1] From wikipedia:
Another disadvantage of LOESS is the fact that it does not produce a regression function that is easily represented by a mathematical formula. This can make it difficult to transfer the results of an analysis to other people. In order to transfer the regression function to another person, they would need the data set and software for LOESS calculations.
There is no formula. Loess is a nonparametric method. It can't be expressed as a simple equation.
You cannot extract the formula from LOESS itself. However, you could simply run another method on the points found by LOESS. If it's a simple 2D graph then it shouldn't be that hard to find a good formula. One method for doing this is symbolic regression (see wiki).
Be aware that this is probably not optimal and it might be better to simply use another method than LOESS.