They surround themselves with talented people.
Influential often deliver speeches in various public spaces (conferences, live streaming online, boardrooms, etc.). Those influential leaders who are wise surround themselves with trusted friends and colleagues who are available on short notice to edit/revise/evaluate their views and/or arguments. These people help leaders think more critically about their messages before delivering them.
They take classes/hire consultants.
In order to gain an edge on the competition, some influential leaders take classes or hire communication professionals to help them refine their skills (in storytelling, developing metaphors, using irony, creating dramatic effect, deploying critical thinking, strengthening impromptu speaking, etc.).
They get feedback from other influential leaders.
One advantage of being an influential leader is access to other influential leaders within the same field. Strategic influential leaders seek out advice from their peers about their ideas. Confiding in people within the same field equips influential leaders with strategies for how to articulate the complexities of a particular product, idea, or trend in ways an audience will find compelling and relevant.
They record themselves speaking
The speeches influential leaders deliver are a key part of their brand. Thus, influential leaders take intentional steps to make sure their messages look, and sound, and feel appropriate for their audiences. One way influential leaders accomplish this is by video recording themselves speaking (both during their rehearsals and during live speaking events). Leaders then study these recordings to evaluate the strength of their delivery, body language, appeal to logic, use of emotion, credibility of evidence, physical appearance, and overall performance.
They are always learning.
One of the most important ways influential leaders improve their speaking skills is by constantly seeking out new and quality information about their field and related areas of interest. They read constantly (and they read good writing!). They study the many ways messages are constructed and communicated (in writing, in speeches, on television, over the radio, on social media, etc.) and they evaluate the strengths and growing edges of such constructions and communications. In short, influential leaders are constantly seeking ways to develop professionally. Consequently, they bring to their speaking a wealth of knowledge about what works and what doesn’t work when presenting ideas publicly.