This phenomenon is called jukujikun in Japanese (熟字訓).
It is when a word's pronunciation is not derived from the standard ON and KUN readings of the individual characters, but actually from the overall meaning of the individual kanji characters. In this case, the meaning of the word is 'delicious', corresponding to 美 (beautiful) 味 (taste). In other words, the characters' KUN and ON readings are completely ignored.
The opposite phenomenon is also possible, i.e. that the meaning of the individual kanji within a word have no relevance to the overall meaning of the word. This is called ateji. A common example is the word 寿司, where the す comes from 寿 and the し comes from 司, but the meaning of these kanji is unrelated to the food. In other words, the characters' meanings are ignored.
These ways of reading kanji are a relic from an earlier time after kanji had recently been borrowed from Chinese. At that time, there were many different experimental ways of reading kanji. Over time, most of these died out due to their complexity, but jukujikun and ateji have remained and are still quite common.
A few other common examples of jukujikun are 大人, 今日, 可笑しい, etc.