The reason that the angles don't change is a little trickier. Here's the idea: There is some triangle DEF, similar to triangle ABC, but with sides twice as long. (Imagine putting ABC in a copy machine and enlarging it by 100%.) The corresponding angles all have the same measures: mA = mD, mB = mE, and mC = mF.
Because triangles are rigid (SSS congruence), any triangles with the same sides as DEF will also have the same angles as DEF.
In particular, the triangle made by doubling the coordinates of A, B, and C has sides twice as long as triangle ABC. That is, the sides are the same as DEF. So the angles are also the same as DEF and as ABC.
That means the sides are in proportion to the sides of ABC (they're twice as long) and the angles have the same measures. So the two triangles are similar.
This argument is what we call general in principle. We explained why doubling the coordinates produces similar triangles. But the same argument would work if you multiplied the coordinates by any number.