How does cut with breaks work in R

cut in your example splits the vector into the following parts: 0-1 (1); 1-2 (2); 2-3 (3); 3-5 (4); 5-7 (5); 7-8 (6); 8-10 (7)

The numbers in brackets are default labels assigned by cut to each bin, based on the breaks values provided.

cut by default is exclusive of the lower range. If you want to change that then you need to specify it in the include.lowest argument.

1. You did not assign labels and default argument in this function is FALSE so an integer vector of level codes (in brackets) is used instead.

2. summary(data1) is a summary of raw data and summary(data1cut) is a summary of your splits.

You can get the split you need using:

data2cut<- 
  cut(data1, breaks = c(1, 3.25, 5.50, 7.75, 10),
      labels = c("1-3.25", "3.25-5.50", "5.50-7.75", "7.75-10"),
      include.lowest = TRUE)

The result is the following:

> data2cut

 [1] 1-3.25    1-3.25    1-3.25    3.25-5.50 3.25-5.50 5.50-7.75 5.50-7.75 7.75-10   7.75-10  
[10] 7.75-10  
Levels: 1-3.25 3.25-5.50 5.50-7.75 7.75-10

I hope it's clear now.