Similarly to how they staff support phone numbers, with teams of people trained in domain-specific knowledge (of the product, account management, etc.). Sometimes these teams are in-house, sometimes they are contracted to a third-party (i.e. how most call centers are).
If you don't think they could handle the volume, that's not really an issue in practice. Chat scales better than phone calls (agents can handle multiple chats concurrently), and just as with phone systems, the chat systems support wait queues. If the queue is long, then you will usually be presented with a contact form or the chat button will just be hidden from view in the first place.
Some of these systems do use bots for initial triage, and the "pro-active" ones that look like an agent is asking if they can help are almost always fake (you only get connected to a real agent if you actually reply).
If you don't think they could handle the volume, that's not really an issue in practice. Chat scales better than phone calls (agents can handle multiple chats concurrently), and just as with phone systems, the chat systems support wait queues. If the queue is long, then you will usually be presented with a contact form or the chat button will just be hidden from view in the first place.
Some of these systems do use bots for initial triage, and the "pro-active" ones that look like an agent is asking if they can help are almost always fake (you only get connected to a real agent if you actually reply).