How much attention should I pay to developments in other research fields?
It depends on your actual research project. Some PhD projects are quite interdisciplinary in the way they are designed (regarding topics, data, method(odology) etc.) while others 'just' talk to their own specific community (although I think, this is rarely the case).  So, to a large extent, it depends on your specific research design of your PhD project how you take into account research from other disciplines. Nevertheless, and based on my experiences as PhD student, I have found it helpful to look into other research disciplines by either talking to people who do research which might be eventually of interest for me or skip through the main journals in the related fields.  For me, (doing my PhD in political science) reading papers from Human Geography, Psychology or Computer Science is quite stimulating from time to time and shows me how other researcher think about issues that I am also interested in. I hope this comment helps you to find a middle ground of helpful interdisciplinarity and getting lost in it. 
PS: I found this blog post helpful, reflecting on this issue: https://www.theguardian.com/higher-education-network/blog/2014/feb/19/interdisciplinary-research-universities-academic-careers