Here you go! A solution in base r.
First we simulate your data, a data.frame with named rows and columns.
You can use sapply() to loop over the column indices. 
A for-loop over the column indices will achieve the same thing.
Finally, save the results in a data.frame however you want.
# Simulate your example data
df <- data.frame(matrix(c(1, 0, 1, 0, 0, 1,
                          0, 1, 1, 0, 0, 0,
                          0, 0, 1, 1, 0, 0,
                          0, 0, 1, 0, 1, 0), nrow = 4, byrow = T))
# Names rows and columns accordingly
names(df) <- c("X", "Y", "Z", "W", "T", "J")
rownames(df) <- c("A", "B","C", "D")
> df
  X Y Z W T J
A 1 0 1 0 0 1
B 0 1 1 0 0 0
C 0 0 1 1 0 0
D 0 0 1 0 1 0
Then we select columns where the sum == 1- columns with unique values.
For every one of these columns, we find the row of this value. 
# Select columns with unique values (if sum of column == 1)
unique.cols <- which(colSums(df) == 1)
# For every one of these columns, select the row where row-value==1
unique.rows <- sapply(unique.cols, function(x) which(df[, x] == 1))
> unique.cols
X Y W T J 
1 2 4 5 6 
> unique.rows
X Y W T J 
1 2 3 4 1
The rows are not named correctly yet (they are still the element named of unique.cols). So we reference the rownames of df to get the rownames.
# Data.frame of unique values
#   Rows and columns in separate columns
df.unique <- data.frame(Cols = unique.cols,
                    Rows = unique.rows,
                    Colnames = names(unique.cols),
                    Rownames = rownames(df)[unique.rows],
                    row.names = NULL)
The result:
df.unique
  Cols Rows Colnames Rownames
1    1    1        X        A
2    2    2        Y        B
3    4    3        W        C
4    5    4        T        D
5    6    1        J        A
Edit:
This is how you could summarise the values per row using dplyr.
library(dplyr)
df.unique %>% group_by(Rownames) %>%
  summarise(paste(Colnames, collapse=", "))
   # A tibble: 4 x 2
  Rownames `paste(Colnames, collapse = ", ")`
  <fct>    <chr>                             
1 A        X, J                              
2 B        Y                                 
3 C        W                                 
4 D        T