I understand why "тарелки" has an "и" at the end as it's plural, but why is "наши" not just "наш" - both mean "our" ?
You need possessive pronoun also in plural, so it matches plural noun. "наш" = singular, "наши" = plural.
As said above, possesive pronouns adjust their gender, case, and singular/plural themselves to the noun they are with. So for example if you want to say "our сountry (страна)" you say "наша страна", страна is nominitive singular female, so наш need to be declined as such.
Sorry for the english :)