What does Priority in Sitemaps exactly do?

I read Google's explanation but I still don't get it.

[Talar]

Lets say somebody searches for the keyword "oranges" in Google. One of my pages has that keyword and Google determines that that is a page it should show in the search results, however that page has a 0.1 priority since I want all visitors to enter my site from the home page, which has a 1.0 priority. Does this mean that Google will list instead my homepage, were the "orange" keyword is not found?

Or, is this only to ensure that in case the crawler is unable to finish crawling all pages, at least it will crawl the most important ones and this is how the chance of those pages being listed is increased?

[mattg3]

This simply states what you deem more important to be crawled in case google won't crawl all.

Hence if google decided it will only crawl 3 pages out of your ten, if you assign high priorities to them they will be crawled and not just randomly. And even if you give a 1.0 to all your pages you will just even the score as then all pages are seen as equal, as far as I understood it.

Theoretically in my opinion you should divide 1 with your pagenumbers and then weigh according to what you think is more important.

so with 10 pages

1/10 = 0.1 for each page. Should you say homepage has 0.2 then one page would need to be 0.

It will have as far as I have understood have an influence on oranges as when google will crawl only 3 pages and the oranges page has a low priority and won't get crawled it won't show up in the serps, if it wasn't indexed at all.

If you deem all content equally important, but don't trust google to do perfect uniform randomisation, you would have to do randomisation yourself with your sitemaps. Hence each day/week etc. give the 0.2 value to another page.

[abates]

Priority in sitemaps has no effect on ranking in Google's search results. It's simply an indication to Googlebot of which pages you regard as important and should therefore be fetched for indexing more often (more often than lower-priority pages on your site, that is).

[annej]

I want all visitors to enter my site from the home page

The world has changed with Google. You will have to assume that people will arrive on all the different pages on your site. Because of this you will need to have good clear navigation especially back to the homepage on every single page on your site.

This is not a bad thing, actually it's a good thing. Read about the value of the long tail. Just search it in Google and you will find a lot of information. The fact is you will get far more visitors to your site because of thousands of specific search phrases that will bring people to various pages on your site.

[Talar]

Priority in sitemaps has no effect on ranking in Google's search results. It's simply an indication to Googlebot of which pages you regard as important and should therefore be fetched for indexing more often (more often than lower-priority pages on your site, that is).

Doesn’t <changefreq> do that?