How do I convince the admissions team that I am passionate about the degree I am applying for?

There are several ways to do this. The best way by far is to demonstrate a specific nugget of interest and knowledge within the (usually) vast field of study you are applying for. So, for example, instead of saying how 'you love English literature', you can say 'the recent resurgence of the supernatural in contemporary literary fiction has really caught my interest' or 'I particularly enjoy Donne's elegies because of their affective sophistication' instead of saying I like Renaissance poetry. The more detail, specificity and depth you can demonstrate, the more effective and interesting your application will be.