Suppose you prove that an upper bound on the integrality gap is $\alpha$.
The goal is to “create” an instance (weighted graph, or better yet, a family of weighted graphs) where the value of the optimal LP solution is $\alpha$
times the value of the ILP solution.
Independent set ILP
$$O_{ILP}=\max\sum_{v\in V}x_{v}\cdot w(v) $$
such that
$$x_{v}+x_{u} \leq 1\ \ \ \forall u,v\in V$$
$$x_{v}\in\{0,1\} \ \ \forall v\in V$$
The relaxation is straight forward. say the value of the optimal solution to the LP is $O_{LP}.$ Now, you want to upper bound $ O_{LP}/O_{ILP}.$
To do this, try to consider a rounding scheme that converts a solution for the LP into a feasible solution for the corresponding ILP. One such scheme involves reasoning about the constraints, that is $x_{v}+x_{u}\leq1$
for every edge in G.
This implies that at least one of $x_{u}$
or $x_{v}$
is atleast $1/2$,
and therefore all vertices in $v\in V$
such that $x_{v}>\frac{1}{2}$
is an independent set in $G$.
I'll leave it to you to reason about how to compare $O_{LP}$
and $O_{ILP}$, (i.e. determine the maximum of the ratio $\alpha=\max O_{LP}/O_{ILP}$), and to find an instance where $O_{LP}$ is atleast $\alpha.$
