The word "total" invokes the totality checker[0] which will report an error if the function doesn't cover all possible cases or cannot be (automatically) proven to not enter an infinite loop.
So, it sounds like if you can't provably account for all possible inputs, the compiler will recognise this and throw an error. How? Magic! (And a solver, feeding your type constraints into the totality checker, I guess...)(Also, bear in mind languages like Haskell use "lazy" or "non-strict" evaluation by default, where values aren't evaluated until they're actually needed; kiiiinda like Streams?)
I'm certainly not an expert on Idris, but I imagine dependent typing provides two benefits:
1) The type signature (which is separate from the implementation; kind of like an Interface in Java) can enforce constraints on your implementation. So, if you define a function with type sig of "Int a => Int b => (a + b)", your implementation must ensure that the returned value is equal to the sum of the two inputs.
2) You can look purely at the type signature to understand what a function is doing. Looking at that type signature above, I can see I want to use that function for addition, not multiplication.
Anyway, I'm absolutely not doing justice to these ideas (mostly because I'm not well acquainted with the languages). If you're curious at all, I would highly recommend taking a course on functional programming as a minimum; once I realised that names are meaningless in code, you start seeing everything as reducible to very similar type signatures, which makes you yearn for a language that lets you write your boilerplate once, in a polymorphic way (beyond generics).