The problem is NP-hard, as proven in Is Minimum Coverage Problem NP-hard?. That answer points out that the problem is called the minimum k-union problem.
When you have a NP-hard problem, you have to find ways to deal with it. One approach is to use an approximation algorithm. In this case, there is a greedy algorithm that does not find the optimal answer but gets within some approximation factor. Follow the links I provided to find it.
Another approach is to use an off-the-shelf solver. In this case, your problem can be formulated as an instance of integer linear programming, and then you can apply any existing ILP solver. To formulate it that way, I suggest introducing a zero-or-one variable $x_i$ per item $i \in S$, and a zero-or-one variable $y_j$ per set $j \in L$. The intended meaning is that $x_i=1$ means that item $i$ is included in set $S'$, and $y_j=1$ means that set $j$ is a subset of $S'$. Add linear inequalities to capture the consistency requirements between these variables: specifically, $x_i \ge y_j$ for each $i,j$ such that item $i$ is contained in set $j$. The requirement that $|S'| \le n$ can be captured by $\sum_i x_i \le n$. Finally, your goal is to maximize the number of sets in $L$ that are a subset of $S'$, i.e., to maximize $\sum_j y_j$. This forms a system of linear inequalities that can be solved with an ILP solver. The running time still might ultimately be exponential, but it lets you take advantage of all of the non-trivial optimizations found in existing ILP solvers.
For your particular problem parameters, I suspect the problem is likely too large to be solved to optimality within any reasonable amount of time. However, many ILP solvers allow you to set a timeout and will produce the best solution found so far within the available time, which might be a reasonable approach to finding the best solution you can for your problem.
