When you start to think of any stream of numbers as a digital signal, a lot of the techniques become useful. For example, user interfaces provide a fairly high-resolution stream of quantifiable user events, like touch positions. Often you want to smooth these in specific ways. If you understand how a low-pass filter works, this is easy.

Does one need a background in electronics to understand DSP?

Not at all. In fact I'd say most DSP these days is done entirely in software.

Any tips on where / how to start? Which books / tutorials worked for you? My background is in software engineering and the only thing I know about DSP is a very very basic understanding of the Fourier transform :)

This book is free online and is a pretty good intro: http://www.dspguide.com/pdfbook.htm