DSLRs never shot videos when they originally came about. All they did was shoot stills, and that too low resolution. But with time comes the demand for changes. People wanted a camera that could shoot videos as well as stills. Thus, the digital SLR became a capable video shooter as well.
At first, it only shot HD. The first DSLR that was capable of doing so was the legendary Nikon D90.
The EOS 5D Mark II, one of the best DSLRs for video shooting, was the first camera that had full HD capability. Modern DSLRs have evolved in leaps and bounds.
Although Nikon was the first to introduce that feature in a DSLR. Canon cameras have traditionally, had features that were geared towards video shooting. But other manufacturers such as Sony are also producing capable video DSLRs.