There are a few things to explain here. First, the function is called glm and I have assigned its value to an object called lrfit (for logistic regression fit). The first argument of the function is a model formula, which defines the response and linear predictor.
With binomial data the response can be either a vector or a matrix with two columns.
If the response is a vector it can be numeric with 0 for failure and 1 for success, or a factor with the first level representing "failure" and all others representing "success". In these cases R generates a vector of ones to represent the binomial denominators.
Alternatively, the response can be a matrix where the first column is the number of "successes" and the second column is the number of "failures". In this case R adds the two columns together to produce the correct binomial denominator.
Because the latter approach is clearly the right one for us I used the function cbind to create a matrix by binding the column vectors containing the numbers using and not using contraception.
Following the special symbol ~ that separates the response from the predictors, we have a standard Wilkinson-Rogers model formula. In this case we are specifying main effects of age, education and wantsMore. Because all three predictors are categorical variables, they are treated automatically as factors, as you can see by inspecting the results: