This puzzle is easier than it looks.
First you have to realise that
In each column and each row, only one cell will be changed. So each changing cell will change its row sum and its column sum by the same amount, and those sums are not changed by any other cells. Therefore the row sums must match the column sums in some order, and their intersections are the cells that have to be changed.
We then simply look at the row and column sums:
I have marked the cells that are in a row and column with an equal sum with hash signs.
250
39 13 24 #25# 68 39 | 208
38 #44# 66 15 41 45 | 249
41 74 41 23 #41# 46 | 266
33 49 41 59 20 #59#| 261
36 43 #21# 58 39 45 | 242
#63# 26 49 28 57 27 | 250
------------------------
250 249 242 208 266 261 249
Now that we know which cells have to change, it is simple matter to change them to make the sums equal to 246. These changes indeed happen to be a permutation of those six numbers.
246
39 13 24 63 68 39 | 246
38 41 66 15 41 45 | 246
41 74 41 23 21 46 | 246
33 49 41 59 20 44 | 246
36 43 25 58 39 45 | 246
59 26 49 28 57 27 | 246
------------------------
246 246 246 246 246 246 246