How are spells ACTUALLY cast?

[Sylvain]

In before Anonymoose the lore master replies

[Anonymoose]

In before Anonymoose the lore master replies

Well you know I won't just walk by my name like that! ♡

how do they actually cast the spells?

I think it's most easily understood with a bit of groundwork.

Spellwork is one type of [aetherial manipulation]. There are many ways to manipulate aether, so there are many ways to "cast a spell", in theory—thanks to the depth of the lore and story/mechanic segregation, some of the nuances become an exercise in semantics. A ninja, for example, also uses hand gestures to manipulate aether. In theory, any ritualized behavior can result in [aetherial manipulation] if the correct mental and technical associations are formed.

That said, almost all the spells you're probably thinking of are indeed, are as RezWolf suspected, ritualized spoken incantation. Conjurer (White Mage), Thaumaturge (Black Mage), Summoner, and Red Mage all explicitly mention incantation / enchantment in some form. Urianger explains that Y'shtola used the dangerous teleportation spell Flow specifically because the safe, traditional spell Teleport was too lengthy.

Though said magicks can also be augmented. For example, many spells rooted in arcanima make use of "arcane geometries"; intricate patterns that map the unlocked mysteries of existence─to draw forth and manipulate the body's aetheric energies, whereby the mage must conjure in his mind the image of distinct mystical diagrams, allowing for increased complexity and precision of spells' effects. Using conductive ink and books enhances and focuses these manipulations (as do most weapon materials associated with a given magical school).

There aren't many limits, just tendencies, really.