In a previous life I used to be a computer programmer, so I can give you the basic idea behind the 'solver' programs...
Working on the basis of the original 3x3 Sudoku grid ...
1. the program, internally, starts off with a 9x9 grid, where each little bit of the grid starts of containing all the numbers 1-9.
2. you enter the numbers that you already have from the original grid, the program deletes all the matching numbers from each row and each column, and each specific 3x3 region.
3. Once you have entered all your starting numbers the program can then display all those locations which have only one possible 'missing' number.
Assuming that the program shows you which is the missing number from a particular row or column, you then enter it back into the grid on the computer, as in number '2' above. Which then leads to number '3', which then leads to number '2', and so on.
That is it, in a nutshell. There is no magical hidden algorithm to help you solve a Sudoku, it is just a few steps repeated time and time again. The same principle holds true for all sizes of Sudoku grid.
There are two kinds of 'solver' program - one which just provides the finished grid, and one which provides you with a clue to a particular number in a particular square as a hint. I prefer/use the 'hint' version.