What is RFormula in simple English?
R formulas are just a common grammar for practical model specifications.

When you input a dataset and a formula like : cancer_probability ~ age + height to a model fitting function. The function will know that it has to work on age and height predictors to provide a model to give a cancer probability output. 

As mentionned in the Spark link you give at the bottom, this grammar rely on some convention, as introducing an intercept by default.