Using RSS clients
A RSS client is a content aggregator, which mean that it gather information from a remote source, the Internet mainly.
RSS exist since 1999 and was adopted by websites as an alternative to newsletters, where a RSS will fetch directly from the website web to get update information. Usually, it is presented by the said website either as RSS or web feed, visually represented by the web feed icon:
Clients
As for me, I use Mozilla Thunderbird.
You can find this strange for reasons, mainly Thunderbird being known as a email client. To that I will say that 1 software to fetch remote content daily, could save the trouble to go on specific application for each thing. Also having it, alongside Mozilla Firefox, both help keeping Gecko browser engine experience consistent and having a long supported software for years to come.
But RSS is client agnostic, so feel free to find out your own client.
Personal note
As you guested it, my website and many on Neocities, do use RSS feed, so feel free to subscribe ;)!