I think people understand the odds are small. However, perhaps they perceive their chances of meaningfully turn around their life in other ways have even smaller odds. i.e. improbable vs actually impossible. At least the lottery doesn't care about your current circumstance and everyone has an equal (equally small) chance.

Secondly, because everyone realizes the chances are small, the real product being sold is Hope. Even the advertisements for the lotteries address this. The thing you're buying is 30 seconds of daydreaming so you can comfortably tackle the rest of the day.

I think: 1) Like you said, people are buying hope. 2) People cannot fathom this degree of improbability. So, the fact that it's at least possible overrides the near-impossibility of it. 3) There is some aspect of entertainment and social-interaction to it. It's a bit like watching sports. Who you're cheering for is irrelevant, and whoever wins doesn't change your life in any way, but still, we watch it.

I know people in the neighbourhood/street I was born, who still live there (well over 50 years in the same house) who bought the postcode lottery tickets since forever, 12x a year, never won anything; this year it fell in the adjacent area code... It must hurt.

Another aspect is that in many states, a large portion of the lottery goes directly into public good programs like education: https://www.powerball.net/distribution-of-revenue

All the players know that the odds are horrible, but in the end someone does win.