Relationship between Transformer Weight and Cooling Method
This big transformers tend to be oil immersed. This oil is mainly used as an insulator, but also helps to cool the transformer. The oil insulates, supresses corona effects and is also the coolant.

In self-oil cooled there's ussually a thermosiphon effect. The hot oil goes up and is circulated through the radiator with no additional parts due to a siphon effect.

Air force-cooling would add a fan to the radiator previously mentioned, but the oil stil circulates through siphoning. So it will be heavier than previously because we have extra parts.

Force oil cooled means that we do not expect the siphon effect to circulate the oil but we add an oil-pump. This oil is then ussually cooled through an oil/water heat exchanger. This means it's heavier still than before with the pump, heat exchanger, etc.

The heavier, the more cooling power you have in this case (and more expensive as well)

Hope it helps.