Strong self-awareness and self-management would let you control your initial impulses or any anxiety you might have around the conversation.
这里的would和might都是情态动词表推测的用法,let you control意思就是“让你控制…”,let的用法是let+sb.+不带to的动词不定式;这句话的意思是:"强烈的自我意识和自我管理能让你控制住你内心的冲动或者是在对话中你可能出现的压力。"