In some places, particularly after cancelling or saving at a save prompt, you will have a small window in which you can move before a cutscene triggers. If you perform a sidehop or backflip in that small window, you can sometimes move outside of the cutscene trigger, thereby skipping it. Two notable uses of this are skipping the Wing Ceremony Cutscene in Skyloft, and skipping a Fi text after the Lanayru Mining Facility.
In general, when collecting items you can skip any associated text boxes by being off the ground when the cutscene would trigger. Some ways of achieving this are:
Different items have different methods that work better for them. In all cases, the text box is not permanently skipped; you have to repeat the text skip each time you want to obtain the item in question.
The easiest way to skip these text boxes is to get close to them, then perform a front hop to collect the item. You can add a jumpslash to improve consistency, but it is not required.
For crystals inside houses, where it is not possible to jump, it is instead possible to both call Fi and grab the crystal at the same time. This skips the the gratitude crystal text as well as the inventory prompt, in exchange with talking to Fi (which is shorter).
When you catch a bug, there is a brief period of time where you can backflip, thus avoiding the text box associated with the bug.
It is not possible to skip the text boxes for these items.