He’s been a lawyer arguing cases and filing amicus briefs w/r/t copyright law and net neutrality, and he’s been an issue-specific activist/lobbyist on those two issues plus, more recently, campaign finance reform. He’s also given a TED talk and written a book outlining zany, politically impossible proposals for tackling the latter issue, and he ran a failed single-issue PAC in 2014.
But that’s a very limited kind of engagement with the overall political system.
If you want to become president, you need to build a broad base of support, which means spending many years organizing, becoming versed in salient political issues, directly working on a wide range of issues with a wide range of other people, leaving a public record and earning credibility. You won’t be able to build a large grass-roots organization, earn endorsements from major institutional political players, build a donor base, etc. on pure message alone. The easiest way to meaningfully engage in a public way is by being elected to political office, but there are probably other possible ways for someone willing to put the years of work in (e.g. as a high-level executive department official, as a career judge, as a military general, ...).
Right now Lessig’s only reputation is as “that guy who doesn’t like copyright, and keeps grandstanding about campaign finance”, but he has no broader credibility as a presidential candidate. Lessig is not “smarter” or “more competent” than the leading candidates; rather, it’s clear that he’s politically naïve in the extreme, has no idea how to run a serious campaign, and would have no idea what to do were he by some miracle elected to high office.