There a some major aspects of stages that tilt the advantage to a certain characters. The most important ones are stage size, blast zones and platforms.
Stage Size
Stage Size is probably the most important metric. Some characters value having lots of space between them and their opponent. Projectile based characters are obvious benefactors; Villager/Isabelle, or the Belmonts are good examples. This gives them time to get their attacks out and allows them the space they need to escape from their opponents and reset the situation. The space also gives them more time to react, allowing them to make better decisions to counter their opponents approach.
On the flip side, aggressive brawlers, such as Fox, prefer to be in their opponents face all the time. A narrow stage reduces the time they need to spend approaching there opponent. Their opponent gets less time to react to them moving. If their opponent does react, the space they have to work with is reduced. They might not the time to use their best option, allowing you to punish them better. Brawlers often need to follow up on punishes, thus a smaller stage reduces the defensive options their opponent has. It is easier to juggle, for example, if your opponent has less airspace to drift in.
Blast Zones
Blast Zones are intuitive. Characters with strong attacks prefer small blast zones. They kill faster this way. It limits how often they need to edge guard by allowing their strong moves to kill outright. Characters with good recoveries are on the other side; they want the extra space so they have an opportunity to recover more often. Characters with bad recoveries will prefer small blast zones since it lessens the importance of recoveries overall.
Platforms
Platforms are the trickiest to explain. Many players prefer having platforms, since it allows them to escape their opponents pressure, or allows them to get the extra height they need to extend combos. On the flip side, you may want to limit your opponents ability to extend combos or dodge your own projectiles. Projectile characters might not like having platforms since it gives their opponents ways to get around their attacks, but it also allows the player to escape their opponents pressure. Its a wash most of the time, since play style heavily influences whether you want platforms. Characters may benefit from platforms yet have disadvantages from those same platforms.
Bringing it Together
Combining these factors together creates small, but compounding factors that influence how a match plays out. This is why Stage discussions are important. Making sure that small stages are also included in stage lists is important to keeping rush down characters viable. The layouts of the platforms is important as well. Layouts like Pokemon Stadium are fundamentally different than Battlefield or Yoshi's Island. This is why diversity is important in stage lists.
What stages you like can be fundamentally different from the character. I hated Dreamland in Melee, despite being a Peach player. I also loved Fountain of Dreams. These were opposed to the consensus for Peach stages, but they worked for me. This is why you should be playing on all legal stages in practice. Its allows you be ready for every stage, including your bad ones that you would usually ban. You might find a pocket counter pick that runs contrary to your opponents expectations.