I was unable to find a suitable answer for the following question:

Once one learns that singular cohomology is the same as cohomology with coefficients in locally constant sheaf, it is natural to try and understand the meaning of LES for relative cohomology $\rightarrow H^i(X,U) \rightarrow H^i(X) \rightarrow H^i(U) \rightarrow H^{i+1}(X,U)\rightarrow$

I am curious to what extent it can be interpreted in terms of sheaves on X using functors $j^*,j_*,j_!$?

It seems that there might be some subtleties if $X\setminus U$ is something like a closed subset in Zariski topology as opposed to "thick" closed subsets...