I'm confused on how this works

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  A'(-6, 12), B'(12, -3), C'(-3, -6)

Step-by-step explanation:

When the center of dilation is not specified, it is presumed to be the origin (0, 0). When that is the case, dilation multiplies every coordinate value by the dilation factor. For example, the x-coordinate of A is -2. The dilation factor is 3, so the x-coordinate of A' is 3(-2) = -6, as shown in your figure.

The same applies to all of the coordinates. (We assume you know how to multiply by 3.)

On the graph, each point moves to 3 times the original distance from the origin, in accordance with the new coordinates. The result is shown in the attachment.