When you have a system of equations, let's say, lines. If you graph them, you'll see that the lines intercept at a common point. This point is the solution of the system. So, when you create another equation based on the equations of your original system, like, adding one to another, you're creating a totally new line, but the solution will be the same.
Think about this system:
$$2x + 4y = 14$$
$$5x + 2y = 19$$
(The system can be viewed as something like this 'matrix')
It has a solution $S\{x=3, y=2\}$
Now, let's say you get the second line
and replace it by the second line minus the first line
, so, you know that the second line is $$5x + 2y = 19$$
Both sides of equation show a mode of writing the number $19$. So, let's say you add $19$ to both sides of equation $1$:
$$2x + 4y + \color{Red}{19} = 14 + \color{Red}{19}$$
Well, I can do it, right? So, $19$ can also be written as $5x + 2y$ (in our system), so let's replace the $19$ of the first side of the equation by $5x + 2y$:
$$2x + 4y + \color{Red}{5x + 2y} = 14 + \color{Red}{19}$$
Now by adding the equal therms we have a third equation to our system:
$$7x + 6y = 33$$
You see? You added one equation to another! :)
That also share the solution $S\{x=3, y=2\}$. Of course, this will be a new line, but in the system of equations you only care about the solutions to the system. It doesn't matters if it will be a different line. You've just found another line that intercepts your system at the solution.
So you changed your system by replacing the second line by the second line plus the first line
If you multiply the two sides of a row, it will still work, because you're getting another equation that also shares the same solution, as I shown here.
When you work in Linear Algebra, you don't solve directly on systems, but I think you're seeing it as something like a matrix. This is just a way to work with systems without having to write the $x, y, ...$ coefficients all the time.
So you had this system of lines:

Then you created a new system with the third equation:

But all them has the same solution