To sum values in data.frame you first need to extract them as a vector.
There are several way to do it:
# $ operatior
x <- people$Weight
x
# [1] 65 70 64
Or using [, ] similar to matrix:
x <- people[, 'Weight']
x
# [1] 65 70 64
Once you have the vector you can use any vector-to-scalar function to aggregate the result:
sum(people[, 'Weight'])
# [1] 199
If you have NA values in your data, you should specify na.rm parameter:
sum(people[, 'Weight'], na.rm = TRUE)