Some recipes, usually cakes, call for the ingredients that we normally store in the fridge—such as eggs and milk—to be at room temperature before using. This is a signal that the batter will be mixed in a fairly precise way, in order to form an emulsion and trap enough air to produce a cake with a tender, melt-in-your-mouth, velvety crumb. If an ingredient that is too cold (or warm for that matter) is dumped into the mix, the emulsion is likely to break. Just as a fragile emulsified sauce, cake batter could curdle and collapse, losing the air whipped into it. The resulting cake may have a dense, compacted crumb instead of the velvety perfection you're anticipating.
In truth, that not all recipes that call for room temperature ingredients really need room temperature ingredients—the room temperature instruction has become a default attachment to recipes that don’t actually require it. But, in absence of being 100% sure, who wants to take the chance of ruining a good cake? So, let's take the instruction seriously.
To get eggs to room temperature, and fast, you can put them, still in their shells, in bowl of warm water for a few minutes, or you can break them into a bowl and set the bowl in a larger pan or bowl of hot tap water. I like the latter method because I can stir—to equalize the temperature—and then check the temperature of the eggs by dipping a finger or using a thermometer. If eggs are in a stainless steel (rather than glass) bowl, it will only take one minute to get three eggs to 70° F.
Be careful—you don’t want to end up with warm eggs, because those can also curdle a batter. Most of us don’t know what 70° F feels like on a finger; it’s actually fairly cool to the touch, so I recommend using a thermometer. You already have one of those for making custards, ice creams, caramel, etc. right? So put it to work. I use the same method for getting cold milk to room temperature, too.