The idea of compactness can not be appreciated unless we see its applications in proving certain non-trivial results. The fact that for a compact set we can replace any open cover by a finite subcover is important precisely because it is far easier to handle a finite number of things as compared to handling an infinite number of things. For instance we can always add a finite number of things (no need to worry for infinite series), or have a guarantee of existence of maximum /minimum of a finite set.
Consider for example that a function $f$ is continuous on a closed interval $[a, b] $. The fact that such a closed interval is compact is a non-trivial result which goes by the name Heine Borel Theorem. Now the function $f$ is bounded locally near each point of the interval $[a, b] $ or to put the matter more formally for each $x\in[a, b] $ we have a neighborhood $I_x$ of $x$ such that $f$ is bounded in $[a, b] $. But from this it is not obvious to conclude that $f$ is bounded on $[a, b] $ precisely because an infinite number of neighborhoods need to be analyzed. But by use of the fact that $[a, b] $ is compact we can just deal with the finite subcover consisting of a finite number of neighborhoods of type $I_x$ and then the function $f$ is bounded in $[a, b] $ because it is bounded in each of these finite number of chosen neighborhoods $I_x$ whose union contains $[a, b] $.
One can use compactness to prove the intermediate value property of continuous functions as well and also the fact that continuous functions attain their maximum/minimum values on a closed interval. There is another result about uniform continuity of continuous functions on a closed interval which does not have a proof without the idea of compactness. All of these applications involve arguments which necessarily require handling a finite number of things. You should try to have a look at such proofs (or better attempt to provide a proof yourself) in order to see how important the idea of compactness is.
You should also pay attention to the proof of Heine Borel Theorem which links it to other forms of completeness of real numbers.