Learn Spanish (Spain): Dejo las llaves en esta cestita de madera. - I leave the keys in this little wooden basket.

It comes from cesta (basket). To add the diminutive suffix, we drop the final vowel and add -ita. Here, it translates to little basket and emphasizes its small physical size.

Would it be wrong to use este instead of esta before cestita?

Yes, it would be incorrect. A word with a diminutive suffix always keeps the gender of the original noun. Since cesta is feminine, cestita is also feminine, so it must take the feminine word for this, which is esta.

Why does the sentence use dejo for leave instead of salgo?

Dejar is used when you are leaving an object behind somewhere, like leaving your keys in a basket. Salir is used when you physically exit or leave a place, like leaving your house.

Why does Spanish use de madera instead of a single adjective for wooden?

In Spanish, what something is made of is almost always expressed using de (of) plus the noun for the material. So, wooden is expressed as de madera (of wood), just like plastic is de plástico.