But how should that gain in energy output be possible without making the whole system huge, too? The heat generating part of a nuclear power station only makes up a small fraction, most of the material is for transforming the heat to electricity and to shield off radiation. If you want to process more heat, you need more infrastructure to do so.

But why would a fusion reactor produce 10 times more power than a fission reactor of the same cost? A fission reactor is, comparatively, very simple. Just a steel cylinder filled with fuel and control rods. No superconducting magnets, no zillion degree plasma to contain and control, low neutron field (as long as we're comparing to D-T fusion). Also heat transfer is much more efficient, enabling high power density, since you pump coolant through the entire reactor vessel instead of just the outer edges.