Let's consider a few terms of the sequence $y$:
$$
y[0] = \frac{x[0]+x[-1]+x[-2]}{3}
$$
$$
y[1] = \frac{x[1]+x[0]+x[-1]}{3}
$$
$$
y[2] = \frac{x[2]+x[1]+x[0]}{3}
$$
Notice how the values of $y$ are always an average of three values. Also, notice how the indices of $x$ "shift" to the right in the expressions, and the next value gets shifted in. This is as if we have a longer sequence, $\{x[-2],x[-1],x[0],x[1],x[2]\}$, and we have a window of three consecutive values of the sequence, and the window shifts over one place to the right for each term in the $y$ sequence.
$$\{\color{red}{x[-2],x[-1],x[0]},x[1],x[2]\}$$
$$\{x[-2],\color{red}{x[-1],x[0],x[1]},x[2]\}$$
$$\{x[-2],x[-1],\color{red}{x[0],x[1],x[2]}\}$$
This is sometimes called a sliding-window average as well because of this property.
As for a concrete example, let's consider the sequence
$$
x[n] = n^2/10, \;\; n\geq 0 \\
x[n] = 0, \;\; n<0
$$
This is plotted below.
![x[n]](https://i.stack.imgur.com/HxhBD.png)
Now, if you do the calculations, you average the first three points, then the second three points, then the third three points, etc. I leave the calculations out, but the result it as follows:

The corresponding numerical values are given in the table below.
n x y
-2 0 0
-1 0 0.0333
0 0 0.1667
1 0.1 0.4667
2 0.4 0.9667
3 0.9 1.6667
4 1.6 2.5667
5 2.5 3.6667
6 3.6 4.9667
7 4.9 6.4667
8 6.4 8.1667
9 8.1 10.0667
10 10 12.1667