Learn Dutch: Dit boompje groeit snel. - This little tree is growing fast.

The root word boom ends in an 'm'. In Dutch, words ending in an 'm' (especially with a long vowel before it) take the suffix -pje to make them a diminutive. Saying boompje is much easier to pronounce than 'boomje'.

Since boom is a de word, why does this sentence use dit instead of deze?

You are right that it is normally de boom. However, all diminutives in Dutch automatically become het words, no matter what their original article was. Because boompje is a het word, we must use the demonstrative pronoun dit (this) instead of deze.

The English translation says 'is growing', but the Dutch sentence just says groeit. Why is there no word for 'is'?

Dutch rarely uses a continuous tense like 'is growing' or 'is doing'. The simple present verb groeit translates to both 'grows' and 'is growing' in English, depending on what makes sense in context.

Does snel change its spelling because it is describing an action here?

No, it stays exactly the same. In Dutch, the adjective form (used for nouns, like 'a fast tree') and the adverb form (used for actions, like 'growing fast') are usually identical. You do not need to add an ending like the English '-ly'.