Why is there a ‘d’ in the word fridge but not in the word refrigerator?

Allow me to write a more general answer.

The phonotactics of a language, and the conventions of its spelling, can lead  speakers to expect letters to be pronounced differently in different contexts—for example, at the start or at the end of a word.

Truncation, in words like (re)frig(erator), takes a sound from the start or middle or end of a word, and makes it a new word. So [ɹɛfrɪdʒəɹeɪda] becomes [frɪdʒ].

But when you come to spelling that truncation, you find that keeping the old spelling can be misleading in its new context (which is after all, a brand new and unfamiliar word). So as both other respondents have written, you can’t spell  [frɪdʒ] as (re)frig(erator) > frig: <g> at the end of a word in English is always [ɡ], so you have to add a final <e>.

Moreover, English typically spells final [dʒ] as <dge>. Plain <ge> does exist, particularly in old –age loans from French; but an unfamiliar word ending in <ige> could be taken for a recent loan from French, and pronounced in the recent French fashion: [ʒ]. Even if that risk didn’t exist, spelling will prefer the usual <dge> pattern anyway, because familiarity in spelling is important (and when deciphering unfamiliar new words, we need all the help we can get).

Australian English truncates words a lot, and has to deal with this issue. The truncation of breakfast can’t be breakie: the shortening of the <ea> is irregular, and wouldn’t be extended to the new word. So it’s usually spelled brekkie. The truncation of poverty can’t be povo, because English shortens long vowels on the third syllable back, not the second; so it is spelled povvo instead, with the double consonant indicating that the first <o> is short. Ditto Seppo as the truncation of Septic Tank = Yank = American.