Neo-Confucianism
Neo-Confucianism (Lǐxué/Dàoxué) is a school of thought, differing from (classical) Confucianism that developed during the Song Dynasty and aims to explain the universe with a mix of Daoist and Buddhist reasoning. Where Confucianism doesn't really get into detail on the afterlife or the soul, Neo-Confucianism aims to rival Daoism and Buddhism, historically to stay relevant.
Why does Neo-Confucianism exist?
During the Tang Dynasty (618–907) and early Song (960–1279), Buddhism and Daoism became extremely popular among populace and government. These philosphies/religions which treaded into new territory such as meditation and metaphysics (explanation of existence) left Confucianism in the dust, and Confucian scholars worked to expand the philosophy so it wouldn't fade into obscurity.
Four Books
While Confucianism already had a foundational canon (most important set of books that define a philosophy), those being the Five Classics, Zhu Xi, important philsopher in Neo-Confucianism, expanded this canon by picking four new books since the original five were important but were obscure, ancient and didin't clearly give an outline of ethical guidance. He wanted texts that were accessible, easy to digest and could rival Buddhist and Daoist texts.
Li
Not to be confused with classical Confucian li, which is ritual. Li, in the context of Neo-Confucianism, is the universal, underlying principle of the universe that governs everything. Sometimes compared to Classical Confucianism's Tian, for example Cheng Yi, one of the leading philosophers who guided Neo-Confucianism said "Tian is Li", though with the difference that Li doesn't have any will.
Qi
Qi is, according to Neo-Confucianism, dynamic sybstance that allows for the existence of the universe. Think of it like the backbone that allows you and everything else to exist. Qi is important, because depending on the state of one's qi it can change their behaviour. For example, balanced qi allows for virtue whereas imbalance causes evil.
Taiji
Taiji is the 'source of all Li' or Supreme Ultimate, which also is responsible for generating Yin and Yang (two opposing forces that cause balance), which in turn produce the Five Phases (forces that create all phenomena, wood fire earth metal and water), Taiji is a creating principle, similar to Li.
Xing is human nature. Mencius, Classical Confucian philosopher, saw human nature as inherently good. Neo-Confucianism agrees, saying that the original nature (benxing) is pure Li, perfection. Physical nature (qizhi zhi xing) is influenced by one's qi and can be clouded by one's desires. Learning and investigating things can help clear up qi so li can be revealed.
Gewu Zhizhi
Gewu Zhizhi, or translating to English "Investigating things to extend knowledge" speaks for itself and the main purpose for doing it is that investigating things can help you understand Li by studying books, rituals, nature among other things. By gradually building up knowledge, you become wise and recieve moral clarity because you are knowledgable.