Its not that they change type. Its that the comparison function returns either a proof that one is greater then the other, or that they are equal. When you are in the right branch, you can pass that proof (type) along with the value into other functions.

here's a full code example:

    import Data.String

    -- takes two integers, and a proof that x < y, and yields an integer
    add : 
      (x : Integer) -> 
      (y : Integer) -> 
      (prf : x < y = True) ->     -- require a proof that that x < y
      Integer
    add x y prf = x + y

    main : IO ()
    main = do
      sx <- getLine -- read string from input
      sy <- getLine -- read string from input
      let Just x = parseInteger sx -- assuming int parse is ok, else error
      let Just y = parseInteger sy -- assuming int parse is ok, else error
      case decEq (x < y) True of   -- decEq constructs a proof if x < y is True
        Yes prf => print (add x y prf)
        No => putStrLn "no prf, x is not less than y"

lets say I mess up the sign of the comparison on the case line and write decEq (x > y) instead... then I'd get a type error

    When checking argument prf to function Main.add:
    Type mismatch between
                x > y = True (Type of prf)
        and
                x < y = True (Expected type)

there's no way to construct the prf value artificially, or sneak in different parameters that are unrelated to the prf value.

it's either a compile error or it's valid.

that is pretty awesome

An existential type - "some unknown types x (a subtype of integer) and y (a subtype of integer) for which LT x y" (or else the other branch). Languages designed for these techniques generally make it easier to write those types than it is in say Java (and in some languages it would be impossible to write that type at all) and infer them so you're not constantly writing them, though there's usually a way to express them directly/explicitly if you need to.