What is the motivation behind defining Deterministic Looping Automata?
In a looping automaton, the transition function is not assumed to be total. That is, for a state $q$ and a letter $\sigma$, it could be the case that $\delta(q,\sigma)$ is undefined.
Intuitively, this is the same as having a transition to a rejecting sink.

So they are "deterministic" in the sense that there is no non-determinism. But still, this means that for every word either there is a single run, or there are no runs. Thus, saying that a word is accepted iff it has a run on it is not a trivial acceptance condition.

The reason for studying these automata is that they correspond to safety properties. A safety property is a language $L$ such that for every word $w\notin L$, there exists some prefix $x$ of $w$ such that for every suffix $y$, $x\cdot y\notin L$. That is, once you read $x$, there is no way to get in the language. 
These properties are useful in practice, as they correspond to "system failure", in a way. That is, once something really bad happens, you can't recover.

Technically, these autoamta are useful because of their simple acceptance condition.

As for references, start here. Also, Orna Kupferman has several papers using these objects (but I suspect you already know that... :) )