The definition can be reformulated as follows - "Let $(X,d)$ be a metric space, a subset $S∈X$ is called compact, if for all infinte sequences $\\{ x_{n}\\}_{n=1}^{∞}\subseteq S$ the following holds: $\\{ x_{n}\\}_{n=1}^{∞}$ has a concentration point and if $\bar{x}$ is a concentration point of $\\{x_{n}\\}_{n=1}^{∞}$ , than $\bar{x} \in S$." Now consider the definition of a concentration point of a infinite sequence. The practical side of compactness of a given set is that it contains it's "edge". If any definition seem vague to you, try to rewrite it using relevant notions with which you're more familiar with.