I think this will work, though leaving you without the expected column names
exp_counts <- read.table(file="mouse_mammary_counts2.txt", sep = "\t", header = FALSE, row.names=1, skip = 1)
Or this will bring in the first column as a normal column and not as row names.
exp_counts <- read.table(file="mouse_mammary_counts2.txt", sep = "\t", header = FALSE, skip = 1)
The column names can be assigned afterwards with the colnames() function.