Beating a lot of air into the "batter" allows the mixture to rise up when baked, as the tiny air bubbles in the batter expand with heat and cause the mixture to grow in size. As the heat increases, the eggs in the batter (if it contains them) begin to set and therefore the mixture doesn't collapse once it cools. The combination of fat and flour also sets the batter (if used in the correct proportion) and again this prevents the mixture from collapsing once it cools.
For this same reason, it's impossible to substitute oils for softened butter in many recipes. Oils can be used successfully in some cakes and muffins, as explained in this column some weeks back now, but it will be impossible to get a light fluffy cake from an oil-based batter. Oils are best used in dense cakes like carrot cake as their purpose is to provide moisture to the cake which means it will keep well without drying out too much.
Another area where the butter's consistency is important is in the making of pastry. There are myriad recipes out there. In some butter is rubbed into the dry ingredients and in others, admittedly fewer, it's melted and stirred in. In puff pastry butter is used in two different ways. The first amount of butter, icy cold, is rubbed into the dry ingredients and used to form a dough. Then a second amount of butter, slightly cooler than at room temperature, is formed into a book-shaped block which is wrapped in the dough. It's rolled out and folded into three, again and again with resting in between. What this means is that eventually, once the process is finished, you have many layers of very thin dough separated by very thin layers of butter. When the pastry is baked the layers of butter expand but are kept separate by the dough - which causes the pastry to rise. The French created a puff pastry dessert called millefeuille (found in many good cake shops) which translates to "a thousand leaves" (or sheets) which is very apt.
So, to reiterate, if the recipe says softened butter then it needs to be of the consistency of margarine rather than oil. It needs to have body and be "spreadable" with a knife. If you think it's a little too soft, place in the fridge for 10 minutes or so to firm it up a little. But first and foremost, make sure you follow the instructions to the dot because they've been written for a reason.
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