Like Japanese and Korean, Vietnam was under the cultural influence of China for a long time and thus started out by using the Chinese script. Unlike Japanese and Korean, however, Vietnamese is not agglutinative — most of its words are one syllable, and there aren’t a lot of changes in the word form (like adding “-s” for plural, for example). This meant that Chinese script worked better for Vietnamese than for Japanese or Korean.
However, Chinese script wasn’t perfect, and as a result, they created their own symbols for some words. If the Vietnamese word had been clearly borrowed from the Chinese, they would just use the Chinese character. (Imagine if French had a logographic writing system that English had borrowed: we would probably use the French character for “chateau” as is.) If the word had come from Chinese, but a long enough ago that it was considered a Vietnamese word, they would use the Chinese character, but with a small tick mark in the upper right. (Continuing the analogy from above, this would be like using the French character for “independence”, but with a tick to show that you should use the English pronunciation.)
Unlike e.g. Khitan, they didn’t just make up new characters for Vietnamese words. For native Vietnamese words, they would either smoosh together two Chinese characters or modify a Chinese character (usually by deleting some of the strokes).
When they smooshed two characters together to make one new character, they sometimes combined two characters for their semantic meaning. For example, Chinese script “heaven”+”above” became one Chu nom symbol for the word that meant both “sky” and “heaven”.
More often, they picked one symbol for meaning and one symbol for sound. The glyph above, for example, combines the sound “ba” with the symbol for “three”, to mean the word “three” which is pronounced “ba” in Vietnamese. Note that the pronunciation syllable frequently used the Cantonese pronunciation and not the Mandarin, as Vietnam is geographically closer to more Cantonese speakers than Mandarin speakers.
The script didn’t fully catch on. Almost all governmental writing was in Chinese script (except for the seven years 1400-1407). A lot of popular literature and poems were written in Chu nom, but writing in Chu nom still required knowing a lot of Chinese script, but most of the more formal writing was still in Chinese script.
Vietnam has also been under the rule of China or the French for significant periods; both occupiers banned Chu nom for some or all of their reign, cutting Vietnam off from large segments of its pop cultural history, and in the case of the French, from their governmental and high literature history as well.