Specifically for the character "兑", the top two strokes are a corrupted form of 八 , which in this case symbolizes air coming out of one's mouth. The character "兑" is an early form of "悅", which means happiness.
Most characters can be classified as Pictographs (象形), Ideograms (指事), Compound-ideograph (會意) or Phono-semantic (形聲). By definition, compound-ideographs are composed by adding two different components to directly infer their meaning. For an example, 林, composed of two 木, which means a forest.
So, attempting to understand Chinese by decomposing them into their respective parts only makes sense for compound-ideograph characters. Unfortunately, compound ideograph characters are only a fraction of the frequent Chinese characters in use. ("兑" is, unfortunately, a pictograph)
Furthermore, many commonly-used Chinese characters take forms and meanings that are many degrees removed from their etymology. Normal people don't learn how to speak and write daily English by learning Latin and Germanic roots, so learning the etymology of Chinese characters would likely be overkill for mastering proficient use of the modern language.
FYI, simply for the sake of correctness, many online sources claiming to explain Chinese character etymology is unreliable and rooted in folk etymology to propagate certain beliefs or religion, such as confucious beliefs or falun gong. For example, many such websites would explain the character "王" by quoting the Confucious saying "a ruler is the person who can connect the heaven, the people and the ground" (一貫三為王).
However, oracle carvings from ~1500BC indicate that "王" is a pictograph for an axe. Confucious is from ~500BC. Chinese characters are far from as "deep" as some propagandish websites would lead you to believe.
But, most native users wouldn't care about the etymology anyway.