What is a "slice" of a parameterized problem $(Q, \kappa)$?
The way you state it, a slice of an parameterized problem is the problem where we use some fixed constant for our problem parameter, instead of a 'free' variable. In a sense, we are only considering the subfamily of the problem for this particular constant, which is probably the logic behind the name 'slice'.

Indeed, if a problem $(Q,\kappa)$ is fixed parameter tractable, there is an algorithm that solves this problem with running time $O(f(n)\cdot g(\kappa))$, where $f$ is (sub)polynomial and $g$ can be superpolynomial. So, if we take a slice of this problem, then there is an algorithm that solves the problem '$(Q,l)$' ($=(Q,\kappa)_l$) in time $O(f(n)\cdot g(l)) = O(f(n))$, as $l$ and therefore $g(l)$ is constant. So $(Q,\kappa)_l$ is indeed solvable in PTIME.