A bit of info ...
Entirely from memory, and dating back a few years to when some of this first "blew up"
IIRC TiVo weren't claiming a patent on "a DVR" but on certain techniques used by most low cost DVRs. In brief, the main patent is on a technique where the supervisory processor takes care of moving blocks of data to and from disk (using DMA on the hard disk controller). The receiver simply dumps data to a block of memory where it's been told to put it by the processor, and the video decoder streams video from the block of memory it's been told to read from.
The key patent is on the method whereby the supervisor processor only keeps track of blocks of data, while hardware takes care of moving/converting the data. For playback, the processor merely has to get some data into memory, then instruct the decoder chip to start reading it - feeding new instructions to the decoder as it eats through the data. If (for example) you skip forward, the processor merely tells the decoder to start again at a new address.
At no time does the processor need to deal with the video/data - only indexes to blocks of data. This means that you only need a low spec (and hence cheap) processor to be able to do some fairly high-end video. Bear in mind that when all this was originally being developed, even "high end" general purpose computers would struggle to do decent video playback.
I took the trouble to read the actual patent a few years ago when there were cries of "Oh no, TiVo has a patent on handling video" - and in particular there was a discussion about where this left MythTV. It doesn't affect MythTV since Myth doesn't use this method of having a low spec processor that merely tells the decoder where to get the data from - the main CPU actually handles the data and feeds it to the GPU (where hardware decoding is used) or does software decode otherwise.
These days, I suspect it wouldn't be hard to build a PVR that didn't infringe on TiVo's patent since even low end processors are quite powerful. But the case in the article will be seeking compensation for all those millions of PVRs already sold.
As an analogy, it's like the difference between a large retailer who handles lots of goods through big warehouses (eg Asda, Amazon) and one who merely operates a marketplace (eg eBay or Amazon marketplace) without actually handling any of the goods itself.