Let $X$ be a smooth variety and $\Sigma$ be a simple normal crossing divisor. Let $i: X\backslash \Sigma \to X$ be the inclusion. Then it is claimed that $$i_{!}\mathbb{C}_{X\backslash \Sigma} \subset \mathcal{O}_{X}(-\Sigma).$$
I guess here $i_{!}$ is the pushforward map $\sum(-1)^iR^ii_*: K_0(X\backslash \Sigma) \to K_0(X)$.
I'm sorry that I don't have any further clue about the meaning of $i_{!}$ here. But I feel an expert in Hodge theory should immediate realize such inclusion. Many thanks for explanations or references!!
There is no $K$-theory in this statement. Only sheaves. The functor $i_!$ is the extension by zero, with the property that $(i_!\mathcal{F})_x=\begin{cases}\mathcal{F}&\text{if $x\in X\setminus\Sigma$}\\0&\text{if $x\in\Sigma$}\end{cases}$.
This sheaf fits into a short exact sequence $0\to i_!\mathcal{F}|_{X\setminus\Sigma}\to\mathcal{F}\to\mathcal{F}|_\Sigma\to 0$. Apply it to the constant sheaf you get a short exact sequence $$0\to i_!\mathbb{C}_{X\setminus\Sigma}\to\mathbb{C}_X\to\mathbb{C}_\Sigma\to 0$$ Now you also have the short exact sequence $$0\to \mathcal{O}_X(-\Sigma)\to\mathcal{O}_X\to \mathcal{O}_\Sigma\to 0$$ And finally, you have inclusions $\mathbb{C}_X\hookrightarrow\mathcal{O}_X$ from the sheaf of locally constant function to the sheaf of regular ones. This holds also for $\Sigma$. Hence a commutative diagram : $$\require{AMScd} \begin{CD} 0@>>> i_!\mathbb{C}_{X\setminus\Sigma}@>>>\mathbb{C}_X@>>>\mathbb{C}_\Sigma@>>>0\\ @.@.@VVV@VVV\\ 0@>>>\mathcal{O}_X(-\Sigma)@>>>\mathcal{O}_X@>>>\mathcal{O}_\Sigma@>>>0 \end{CD} $$ By universal property of kernel, you have an inclusion $i_!\mathbb{C}_{X\setminus\Sigma}\hookrightarrow\mathcal{O}_X(-\Sigma)$.