This is a case of liaison: in certain cases, when a word whose spelling ends with a silent consonant is followed by a word whose pronunciation starts with a vowel sound, the final consonant becomes pronounced as if it was the first consonant of the second word.

I’m not going to list all cases where liaison is mandatory, possible or impossible here. The general idea is that there is only a liaison if there is a sufficiently strong connection between the two words, but this is far from covering all cases. Here, we are in the specific case of an adjective followed by a noun, and the liaison is almost always compulsory. (On the other hand, there is never a liaison between a noun and a following adjective.)

Normally the liaison sound simply consists of taking the consonant at the end of the first word and pretending that it was added at the beginning of the second word. However, in the case of a singular adjective followed by a noun, the pronounciation of the adjective can change. The general rule is that the adjective is pronounced as if it was the feminine form, which can modify the last vowel sound.

Un bon ami: [bɔ̃] + [a.mi] → [bɔ.n‿a.mi] (as bonne + ami)
En plein air: [plɛ̃] + [ɛʁ] → [plɛ.n‿ɛʁ] (as pleine + air)

(In phonetic notation, . indicates a syllable boundary; ‿ indicates a liaison and the pronunciation is the same as if the ‿ symbol was not present.)

This rule also works in cases where it’s the normal liaison sound.

En piteux état: [pi.tø] + [e.ta] → [pi.tø.z‿e.ta]
Un long entretien: [lɔ̃] + [ɑ̃.tʁə.tjɛ̃] → [lɔ̃.g‿ɑ̃.tʁə.tjɛ̃]

But there are a few exceptions.

Un grand homme: [gʁɑ̃] + [ɔm] → [gʁɑ̃.t‿ɔm] (as if it was spelled *grant)

In a few cases, the spelling of the adjective even changes to reflect the different pronunciation.

Un vieil homme (un homme qui est vieux).
Un bel homme (un homme qui est beau).
Un nouvel homme (un homme qui est nouveau).

This adjustment of the first word only happens for singular adjectives. For example, contrast:

Un bon ami: [bɔ̃] + [a.mi] → [bɔ.n‿a.mi] (as bonne + ami)
Mon ami: [mɔ̃] + [a.mi] → [mɔ̃.n‿a.mi] (as mon + n + ami)
De bons amis: [bɔ̃] + [a.mi] → [bɔ.z‿a.mi] (as bon + z + amis)