When playing Yu-Gi-Oh what happens when you add a level to a monster

Nothing usually. If somehow you added a level to it in hand, then it may now require tribute, or more tribute than before.

On the field, a level change only affects things that directly refer to level. Fortune Ladies have ATKs based on their level, so increasing theirs is good for them. As well as this, Synchro Summoning looks at the monsters' level, being able to add levels to a monster might change your Synchro Summoning options.