I just don't know why a staff would have any impact on for example my mental stats, such as Intelligence,
Well, maybe not "intelligence". However, the staff may have an effect on how useful intelligence ends up being. So "intelligence usability" is what goes up. We just typically abbreviate that as "intelligence" as no distinction is usually needed.
Some of these apparent boosts may just be deception, much like other forms of magic. Maybe the wielder doesn't actually think with more clarity or focus, but that is just the reputation a staff may have, and that reputation makes results seem more remarkable, so the staff's resulting power ends up actually being more effective even if it is all false.
explain the increase in magical power by equipping a better staff. This also means I don't know how to narrate what a better staff is. Is it lighter? Heavier? Fancier? Mundane?
What you'd be looking for is a staff that is more magical.
What is more magical to you?
A local McDonald's restaurant?
A toy store that causes children to be happy?
Or an entire theme park like Walt Disney World?
Think about this from the eyes of a child, when such "magic" hasn't yet been dulled by finances. Which of these places seems magical to you?
From what would you expect more magic to come?
A magician that pulls off his top hat, and is wearing white gloves, right by a sign that says "Forgot my wand, but I will do my best anyway", and a deck of cards on a table
A magician who did not forget his foot-long (30-centimeter-ish) black rod with white caps on both ends, and who waves the wand around
A witch, complete with pointed hat and wart, standing by a cauldron that has steam coming out of it, cackling as she speaks
A six foot (2 meter) tall wizard with a long beard, pointed hat, and a staff that is seven-and-a-half feet tall, with a glowing red gem
See, even you, non-magic user, have some idea of just what is magical.
Rarity can also help a whole lot. A shiny rock is better than a common stone, but a semi-transparent gem is better, and a rock that reflects and amplifies light is even better. A harness made of gold will glitter more magically than black iron.
Reputation can also be good. A story can multiply its usefulness. Having a staff which has lasted a remarkably long time can be good. Even better is a weapon that has had songs written about it. If that's not feasible, at least strong rumors that it was used by a famous legendary figure from the past may suffice. Preferably if the item has been unused (and especially if out of sight) for days. No, actually, millennia would be even better. Oh, and try to throw in a name of an unvisitable location or three.
The worlds of magic differ from the physical world in which we interact. Our bestest of technology relies upon electricity that is generated, stored, or transported from a source using conductive material. Magical power relies on similar such fuel or components, fictitiously in the forms of swirling lights, incomprehensible mutterings, and special ingredients, along with non-fictitious sources such as preparation, superior knowledge, concealment, and distraction.
One of the strongest magical powers is awe. If you take a 4-4 piece of wood, it may have no magic. However, carve away some of that wood to reveal the image of a gargoyle's head at the top of the staff, and suddenly you have magic.
The world of magic also relies on such things as blessings, hopes, dreams, secrets, hexes, and curses. Words common in magical stories involve fanciful creatures, myths, legends, and other epic tales involving incomprehensible elements such as unfamiliar abilities to exaggerate and distort normalcy into illusions that are completely unverifiable. Most importantly, lots of unfamiliarity is helpful to generate further awe. Using inhuman supernatural resources might be one method of generating such unfamiliarity, but so can easier methods like just using skills from a different culture or social circle.
Elimination of the ability to use one's right mind, whether from foreign substance or disease or simply sleep deprivation, can be helpful.
Less commonly reported, but possibly mysterious all the same, is if the "staff" transforms. Having it appropriately match the owner's wardrobe may also help to strike fear in enemies.
So there's lots of ideas. Of course, there's certainly the possibility of combining multiple of these elements. You can also combine other elements. Tapping other powers, like the natural powers of biology, elemental powers, and even chemistry, can be very great as long as they remain unfamiliar, or at least uncontrollable. (Less familiar powers, like those involving dimensional rifts and physics belonging to other worlds, may be even better.)
Even science, like being able to use a "hang glider" to move about (though only downward) through the air, can be effectively utilized magically if the verifiable, scientific element can be effectively cloaked. (What is required for "effective" cloaking may vary depending on your audience.) And if you can get trustworthy but wrong audiences to make their stories more interesting, you may even be able to upgrade your reported abilities, like upgrading from gliding to flight.
Especially if the tales are old, in which case they may "reboot" and make things even more over-the-top.
Is it lighter? Heavier? Fancier?
Fancier? Yes. Lighter? Heavier? Whatever is more magnificent, which may vary based on how well it matches the user(s), and what environment it is in. You might think that the strength of a legend would have little effect, but such mistaken belief would indicate a lack of much exposure to magic.
And if this answer didn't seem very helpful because you were hoping for a more tangible, realistic answer, then let me remind you of how real magic really works: unfamiliarity, including concealment which often comes from successful distraction. See, all these things I was writing about actually are exactly what makes a magical staff, amulet, or other thing special.