1. Establish Credibility
Place yourself in the shoes of an influential world leader, and imagine that you're invited to a party by the party organizer, whom you don't know. The obvious answer is no: while the party attendees would receive tremendous benefit, there is little to no benefit for you to come to the party.
Now, imagine you're invited to the same party, but you are informed that other world leaders will be attending, and the party is sponsored by a well-known international conference group. In the second scenario, you're much more likely to attend: the partygoers would receive the same benefit, but now there is a greater incentive for you to attend.
For any event where higher caliber speakers are on the invite list, ensure that you have credibility behind you and that the benefits of speaking are great enough to cause them to consider your offer. The TEDx brand is well-known worldwide for quality conferences (and for good reason, the application process and regulations for TEDx events are stringent), which established instant credibility for my event.
2. Research, Research, Research
Don't just blindly invite big-name speakers to speak at your event, even if you're confident they'll attend ― take the time to thoroughly research them and determine if they'll be a good fit.
I researched some of the speakers for TEDxYouth@BriarWoodsHS for years to ensure their areas of expertise and speaking style aligned with the event, and while most organizers may only need months to vet their speakers, going the extra mile with speaker selection will pay dividends. If there's any place to put in the most effort, it's here: if you invite speakers that don't mesh well with your event, you have to work (and live) with that mistake.
Plus, having a strong knowledge base about your speakers will help you interact with them better and give you more insight into how to work them into the event program. Have a speaker that tends to give emotional, heartfelt talks? Slide them in at the end of the event to tug at the audience's heartstrings before they leave. Have a presenter that prefers strong speeches with a call to action? Put them at the beginning to draw listeners in.
3. Send the Invites
Now that you're certain your preferred speakers are a great fit for your event, you need to do more research to determine how to convince them to accept your invitation. You should have plenty of information from your previous research to draw on when crafting invitations, but now is the time to focus more on what their personal goals are and craft connections between them and your event.
One key thing to note: for some speakers (especially those who deliver talks professionally on a routine basis), it's better to reach out to them through an intermediary (think secretary, manager, or mutual friend), but others prefer being contacted directly through a cold call or email. There's no formula to this, as it varies from person to person, so use your best judgement.
If you do decide to reach out to your speaker through an intermediary, try to establish a relationship with the intermediary first ― it might drastically increase your chances of them forwarding your request to the speaker. Even if the speaker declines your offer, you'll at least have a new contact you can call back in the future.