I'm Jesse Talavera-Greenberg, and I approve of this message.
If I were running for US president, would I have to say that in my televised campaign ads? If so, why?
You don't need that exact statement, but per 52 USC § 30120(d)(1)(B) you do need to have
a statement that identifies the candidate and states that the candidate has approved the communication. Such statement—
(i) shall be conveyed by—
(I) an unobscured, full-screen view of the candidate making the statement, or
(II) the candidate in voice-over, accompanied by a clearly identifiable photographic or similar image of the candidate; and
(ii) shall also appear in writing at the end of the communication in a clearly readable manner with a reasonable degree of color contrast between the background and the printed statement, for a period of at least 4 seconds.
This requirement exists so that people know who is behind slimy ads. If you want to run an attack ad against your opponent, you have to be willing to put your face on it and stand behind whatever you said. It was a lot less effective than was hoped (independent ads have since become a much more prominent part of the political landscape, so candidates don't need to get their hands dirty because supportive super PACs can do it for them), but that was the idea.
This falls under the requirements for financial disclosures. Every campaign communication is funded by someone and the intent of this notice is transparency in campaign finance. Whether a letter, a yard sign, or a TV ad, there must be language that clearly identifies the person or organization responsible for funding it.