there's a website that catalogues and compiles polls from various different sources. each poll asks something like 1500~ people. they do both telephone and online polling, depending on the polling organization. if you click on the poll, you can find out more about the organization and they often publish exact methods and data so you can look for yourself how they gathered the data
now, you're right that the sample size is going to be different than the population. however, there is a science and math to this stuff where you can use formulas in order to account for that. let me give a simple example
let's say you live in Townsville with a population of 60 people. 20 of those people are male and 40 are female. you want to find out whether everyone likes vanilla or chocolate ice cream, so you go to the bowling alley. at the bowling alley, there are 10 men and 10 women. so you survey everyone but you realize
the sample size demographics are different than the actual population demographics. in the population, females outnumber males 2 to 1 whereas in the sample population it's 1:1. so you need to weigh your votes accordingly
you can either do one of two things - you can count every vote from a woman twice. or you can count every vote for a men at a ratio of 50%. that way you are representing the population demographics more accurately
polling agencies do this but with a myriad of different demographic properties. age, sex, gender, income, ethnicity, etc, in order to try to get a more accurate number. you will never be able to exactly represent a population with a small sample size, but you can get pretty damn close within a margin of error.
tldr: polls are not perfect but they absolutely can help predict public sentiment because of some statistical axioms (Law of Large Numbers, Central Limit Theorem, Random Sampling)