Without actual data, it is difficult to show you the exact code you need to use, but after having a glimpse at that png, I would suggest you try something along the following lines: 
library(reshape2)
library(ggplot2)
df <- melt(your_data)
ggplot(df, aes(x=variable, y=value)) + geom_boxplot()
This code probably needs some adjustment. If it doesn't work and the adjustments are not obvious, please post some example data in a way that makes it easy for us to use it. Data from a screenshot would imply we have to manually copy-paste each and every number, which few would be willing to do.
To clarify the general procedure: melt "stacks" all your columns on top of each other and adds a variable called variable, which refers to the old column name. You can hand this over to ggplot and say that the different values of variable should be on the x axis, which is what you want. For instance, have a look at women:
head(women)
  height weight
1     58    115
2     59    117
3     60    120
4     61    123
5     62    126
6     63    129
str(women)
'data.frame':   15 obs. of  2 variables:
 $ height: num  58 59 60 61 62 63 64 65 66 67 ...
 $ weight: num  115 117 120 123 126 129 132 135 139 142 ...
You see that women is a dataframe with 15 observations and two columns, height and weight.
Now, let's melt them:
df <- melt(women)
head(df)
  variable value
1   height    58
2   height    59
3   height    60
4   height    61
5   height    62
6   height    63
str(df)
'data.frame':   30 obs. of  2 variables:
 $ variable: Factor w/ 2 levels "height","weight": 1 1 1 1 1 1 1 1 1 1 ...
 $ value   : num  58 59 60 61 62 63 64 65 66 67 ...
Now you see it has 30 observations, and two columns: variable and value. variable identifies the old columns.
Let's hand this over to ggplot:
ggplot(df, aes(x=variable, y=value)) + geom_boxplot()
yields:

Here you have boxplots for both columns in the original women dataset.