Astoundingly, you
don't if you wait for the live broadcasts to finish and just watch the recorded versions!
A friend got rid of her TV when we went digital here and told me that she was going to just use iPlayer so she wouldn't need a licence. I told her that she was wrong. She insisted that she was right, so I looked it up -
here. She was right!
iPlayer is an example of technology developed in house at the BBC that works. As opposed to the 5-year Siemens/BBC "initiative" that cost them £90 odd million alone for the 2 years the BBC handled...
Due to their public service remit, they gave the technology to ITV rather than charge for it (& C4 I think), which I think is why you can watch live ITV through iPlayer too. The public service remit often means the BBC don't complain about own content programs being put on youtube for example, beyond their control (there might be technical copyright issues regarding footage they don't own).
Anyway, currently there is not a problem with people not paying the licence fee and watching delayed content on iPlayer catch up - online viewing figures aren't yet significant. The majority of BBC viewers are in fact elderly and not tech-savvy, so it probably won't change very fast. iPlayer at least gives accurate realtime online viewing data, so they'll quickly have an idea if sidestepping the licence fee to view this way becomes more of a problem. I'm sure the BBC would look to close the loophole by way of registered TV licence data. If they tried to change the licensing requirement to cover a window beyond the initial broadcast date, I speculate they might need an act of parliament.
Edit: and re the TV licence letters, I think the catchers know well the financial equation. 95% plus of inhabited properties have a TV. If a licence expires/property changes hands:
1) fire off a few generic vaguely threatening letters - most people are worried at this point and buy a licence. Those letters have cost less than £1 to send so job done.
2) if there is no licence bought or response, the licence men show up and knock on your door to get you that way but as others have said, the days of TV detector vans seem long gone.