The grammar explanation
The question word, the interrogative mikä : what is an interrogative pronoun. All pronouns together with nouns and adjectives form a group called nominals, which all are inflected in the grammatical cases. The genitive case is the general "glue" case in Finnish, the genitive form of mikä is minkä (both in sg. and pl.).
The practical explanation
Think this way. The size is a property some object has. Talo on ison kokoinen : (lit.) The house is of big size. In a question you use the interrogative word as the "size", minkä kokoinen?.
"mikä" as a question refers to "which of these". It is singular and refers to only one thing, "mitkä" would be the plural. It can also sometimes be translated into "what".
"Mikä on vikana?" -> "what is the problem"
"Mikä" can also be used as part of a subordinate clause.
"Hän on kuuro, mikä selittää miksi hän ei kuule minua" -> "they are deaf, which would explain why they can't hear me"
If we swap "mikä" and "minkä" in the previous sentence we'd have this.
"hän on kuuro, minkä vuoksi hän ei kuule minua" -> "they are deaf, which is the reason they can't hear me".
"Minkä" generally is used as a sort of comparison when using it as a question. "Minkälainen" "what kind of (compared to what)?", "minkäkokoinen" "compared to what size?" etc.