I translated it like you too, because "This food is not tasty" sounded more natural to me than "This food has no taste." But I think the authors decided to use a more literal translation to teach the "yok"-construction, which is mostly translated as "there is no" or "has/have no".
Because that is how we make "have" sentences in Turkish.
Literally, it's something like "the food's taste does not exist" -- and "the food's taste" is yemeğin tadı.
Turkish marks possession both on the possessor and the possessed, so literally, it's something like "the food (yemeğin) its taste (tadı)".
The -ı on tadı is the possessive ending for "his/her/its taste".
That's how Turkish makes noun-noun compounds. English can simply put two nouns next to each other ("police station") or glue them together into a single word ("bedroom"); Turkish adds a possessive marker to the second one in this sort of situation.
yatak odası is a bedroom; yatağın odası would be "the bed's room".
Some of those noun-noun compounds later become single words but still have that ending, e.g. ayakkabı "shoe" or havalimanı "airport", from ayak kabı "foot container" or hava limanı "air port".
And some of them even fuse so tightly together that it's not treated as a compound word at all -- thus ayakkabı has the plural ayakkabılar. (By contrast, havalimanı has plural havalimanları, just as if it was still two words, hava limanları, with a plural possessive ending on the second word.)