I don't know how Talk Talk has conducted their Teamviewer connections in the past, but here's an example how the intruder might have gained access.
Normally, upon installation Teamviewer creates a unique ID and a 4 digit password that changes every time the TV application is restarted. (The ID can't be easily changed by the end user since it is generated from a MAC address)
If Talk Talk has a) customized the Teamviewer application to never randomize the password and b) enabled the host module to start at boot, and c) Talk Talk customer support has written down the ID and the password in the breached customer records then it is trivial for the hackers to invade computers without any user action.
The above is dependent on several conditions but I've seen worse decisions when managers are contemplating between ease of use and security. Is it possible that someone at TT has made those decisions? Yes.