For example: If a really painful Carbon tax was introduced over night, the next day people's lives would look a lot different. Activities that used to be mundane would suddenly be expensive luxuries.
However, that's ignoring the fact that the economy would start to route around this tax. What are currently still niche low-carbon technologies will get a lot cheaper due to economies of scale. Photovoltaics are already a super cheap way of making electricity while the sun shines - suddenly there's going to be a huge incentive to building out storage.
(Of course, you wouldn't actually do that over night.)
But again, the OP is about credibility. So, for the advocates of legislation changes (e.g. climate activists) it would give their credibility a huge boost if they started to surrender symbolically from things like traveling abroad (and also, would remove a rhetorically very easy target from their critics).
The proposed legislation itself should of course refrain from being "just" symbolic.
If you ARE in fact hurting yourself too much, then you prove that people can chose one from the following:
A) keep the status quo, but the planet will most probably heat up and very bad things happen.
B) change the status quo to something that really, really hurts. But the reward for your surrender is that there is a chance that the planet does not heat up so much and less bad things happen a few decades from now.
That is what most people perceive in climate activists.