Since you don't state otherwise, this answer assumes the stDte and edDte columns are both of "Date" class.
In base R you could use Map() to create the sequence of dates, then data.frame to bring the new data frame together after creating the new Name column with rep.int().
M <- Map(seq, df$stDte, df$edDte, by = "month")
df2 <- data.frame(
Name = rep.int(df$Name, vapply(M, length, 1L)),
Dte = do.call(c, M)
)
str(df2)
# 'data.frame': 65 obs. of 2 variables:
# $ Name: Factor w/ 2 levels "A","B": 1 1 1 1 1 1 1 1 1 1 ...
# $ Dte : Date, format: "2010-05-01" "2010-06-01" ...
head(df2, 3)
# Name Dte
# 1 A 2010-05-01
# 2 A 2010-06-01
# 3 A 2010-07-01
tail(df2, 3)
# Name Dte
# 63 B 2013-12-01
# 64 B 2014-01-01
# 65 B 2014-02-01
Or you can use the data.table package and do
library(data.table)
setDT(df)[, .(Dte = seq(stDte, edDte, by = "month")), by = Name]
