That threw me off as well but it checks out.

An interval includes the starting note in its name so e.g. c to g is a 5th because there are 5 notes in it: cdefg. But if you move up a 5th from c you move by 4 steps: c->d, d->e, e->f, f->g.

In the first example we move a 5th and a 4th. So if we start on c we go 4 steps c->g and 3 steps g->c for a total of 7 steps c->c. An interval that represents 7 steps has 8 notes because we count the starting note as well so it is an 8th.

For the scond example we move an 8th and another 8th. So starting on c again we move 7 steps c1->c2 and 7 steps again c2->c3 for a total of 14 steps c1->c3. An interval with 14 steps has 15 notes including the starting one so it is a 15th.