Given a probability space $(\Omega$, $\mathcal{F}$, $\mathbb{P})$, let $Y$ be a random variable defined on it such that $Y\in \mathcal{L}^1$ and $M_n=\mathbb{E}\{Y|\mathcal{F}_n\}$ be a martingale. For $c>0$ one has \begin{equation} M_n1_{\{|M_n|\geq c\}}=\mathbb{E}\{Y1_{\{|M_n|\geq c\}}|\mathcal{F}_n\} \end{equation} since $\{|M_n|\geq c\} \in \mathcal{F}_n$.
At this point, if one states that:
hence, for any $d>0$, one gets \begin{equation*} \begin{split} \mathbb{E}\{|M_n|1_{\{|M_n|\geq c\}}\} &\leq \mathbb{E}\{|Y|1_{\{|M_n|\geq c\}}\} \\ &\color{red}{\leq}\mathbb{E}\{|Y|1_{\{|Y|>d\}}\}+d\mathbb{P}(|M_n|\geq c) \end{split} \end{equation*}
why does the second inequality (the one in $\color{red}{red}$) hold true? That is, how can I be sure that it is always true, for any pair $d,c > 0$ that $\mathbb{E}\{|Y|1_{\{|M_n|\geq c\}}\}\leq\mathbb{E}\{|Y|1_{\{|Y|>d\}}\}+d\mathbb{P}(|M_n|\geq c)$. Clearly $\mathbb{P}(|M_n|\geq c)$ is nonnegative, but I cannot see a clear relationship between the set $\{|M_n|\geq c\}$ and the set $\{|Y|>d\}$ if one does not know if there is any relationship between $c$ and $d$.
You have that $$\mathbb{E}[|Y|1_{\{|M_n|\geq c\}}] = \mathbb{E}[|Y|1_{\{|M_n|\geq c\}}1_{\{|Y| > d\}}] + \mathbb{E}[|Y|1_{\{|M_n|\geq c\}}1_{\{|Y| \leq d\}}]$$
The first term on the right is bounded above by the first term in your desired inequality since $|Y|1_{\{|M_n|\geq c\}}1_{\{|Y| > d\}} \leq |Y|1_{\{|Y| > d\}}.$ The second term is similarly bounded since $$\mathbb{E}[|Y|1_{\{|M_n|\geq c\}}1_{\{|Y| \leq d\}}] \leq \mathbb{E}[d 1_{\{|M_n|\geq c\}}1_{\{|Y| \leq d\}}] \leq \mathbb{E}[d 1_{\{|M_n|\geq c\}}] \leq d \mathbb{P}(|M_n| \geq c)$$