Try by (?by) if you want base R. If you start doing more complicated things, the plyr/dplyr packages are pretty amazing, and if you are going to muck around with huge datasets and don't mind a bit more of an initial learning curve, the data.table package is also amazing.
A reproducible example would be fantastic.
E.g.
set.seed(1) # so your random numbers are the same as mine
pincome <- data.frame(incomechng = runif(20, min=-1, max=3))
# what you had was fine too; using ?cut is another way to do it
# have just put it in for demonstration purposes.
# though `cut` uses intervals like (a, b] or [a, b) whereas yours
#  are (-Inf, 0] (0, 1) [1, Inf) which is a little different.    
pincome$income_growth <- cut(pincome$incomechng,
                             breaks=c(-Inf, 0, 1, Inf),
                             labels=paste("level", 1:3))
Now we can take the average within each group. I've shown three options; I'm sure there are more.
# base R ?by
by(pincome$incomechng, pincome$income_growth, mean)
# pincome$income_growth: level 1
# [1] -0.6848674
# ------------------------------------------
# pincome$income_growth: level 2
# [1] 0.4132334
# ------------------------------------------
# pincome$income_growth: level 3
# [1] 1.772039
# plyr (dplyr has pipe syntax you may prefer but is otherwise the same)
library(plyr)
ddply(pincome, .(income_growth), summarize, avgIncomeGrowth=mean(incomechng))
#   income_growth avgIncomeGrowth
# 1       level 1      -0.6848674
# 2       level 2       0.4132334
# 3       level 3       1.7720395
# data.table
library(data.table)
setDT(pincome)
pincome[, list(avgIncomeGrowth=mean(incomechng)), by=income_growth]
#    income_growth avgIncomeGrowth
# 1:       level 2       0.4132334
# 2:       level 3       1.7720395
# 3:       level 1      -0.6848674