Taking a broad approach to "affected", this CVE theoretically allows someone to run arbitrary code on a vulnerable host. That means, under the right conditions, information you have placed on remote servers could be accessed by unauthorized third-parties. An easy example could be someone exploiting this vulnerability to exfiltrate passwords or other secrets from the service, but your imagination is the limit.
While large services like Twitter, Apple, etc. may on the surface appear susceptible to this, there are still many steps between making a DNS request (as pictured in the link) and causing real havoc. Things like outbound firewall rules could allow DNS but prevent other protocols. Ultimately we will need to wait and see, service-by-service, what they share about their experience with this vulnerability.
Can someone run code on my machine when I visit Apple websites or Twitter websites? Can I avoid any problems not visiting these websites and services?
Not necessarily, no, but certainly an attacker could theoretically alter the behaviour of the website in a way that changed its behaviour in your browser/app. Certainly limiting your usage of these sites could, again theoretically, reduce risk/reduce the likelihood of your data being readily available to an attacker. That said, there is no reason to believe that services as large as those examples don't have other contingencies in place.