Make up an easy equation that has more than one solution:

x + y = 10

That has lots of solutions because 

1 + 9 = 10, 2 + 8 = 10, 5 + 5 = 10, 6 + 4 = 10, etc., etc.

Now multiply that equation through by some easy number, say, -1:

-x - y = -10

That one has the same solutions that the first one does.

So this system of equation has many solutions:



Now make it into an augmented matrix

Answer:  Because we picked an equation that has many solutions.
Then we multiplied it through to get another equation that 
has the same solutions.

Also if we add row 1 to row 2, we get the
row-reduced echelon form:



And we have learned that when the augmented matrix is
in row reduced echelon form and the bottom row has all
0s, then the solution is not unique. The matrix is
equivalent to the equation:

x + y = 10

and solving for x we get

x = 10 - y

So the general solution is

(10-y, y)

We can substitute any number for y and get a different
solution.

Edwin