The animals don't go: they arrive, in vast, uncountable millions. Beneath the dry salt crust, there are unimaginable numbers of organisms: shrimps, yabbies (freswater crayfish for our over-the-water friends, though "fresh" isn't really the term for this water), fish, insects, you name it. These creatures exist more-or-less in stasis through the dry years, then when the nutrient-rich brown, muddy water comes down the Cooper, they burst forth and multiply at an incredible pace.
Within weeks, birds are arriving from all over the continent: cormorants, waders of all kinds, Black-winged Stilts, the mysterious, endemic Banded Stilt (a listed threatened species, if my memory is to be trusted), avocets, pelicans, egrets, herons, swans, all kinds of ducks .... vast numbers of birds.
Lake Eyre and similar, smaller inland lakes are vital breeding grounds for these birds. How they find the particular lake system that is flooded at any particular time is a mystery unknown to science, but find it they do, and they immediately start breeding while the food supply is plentiful. As the fish and other creatures grow bigger, the lake dries out, and the feeding frenzy increases. Eventually, the water disappears, the birds move off elsewhere, and, somewhere underneath the mud, another generation of small creatures hibernates, waiting for the next big wet.
Lake Eyre is one of the natural wonders of the world.
Now, let's deal with some misconceptions. First, it does not fill up "twice a century"; it fills all the way up around twice a century. It fills to a lesser extent considerably more often than that. There was water flowing into the Lake Eyre system as recently as a few months ago, for example. It would all be gone by now, no doubt, but it is nothing at all unusual to have a little water in the lake.
Second: it is a gross misunderstanding of the natural cycle to think "empty = good" or "empty = bad". Like most natural systems, the Lake Eyre ecology thrives on change, on the filling, drying, waiting, filling again cycle. To disrupt that would be a crime of the first order.
Third: Time has demonstrated the immense scale of the proposed task - so large an engineering task as to be impractical.
Last, and perhaps most important: let us return to the fundamental point: it wouldn't increase rainfall anyway. There is no observed relationship between rainfall in the surrounding countryside and water levels in Lake Eyre - just like Spencer Gulf, it can (when full) produce evaporation, but that moisture doesn't fall anywhere useful. We already know this from examining rainfall records in years when the lake was full.
If you really want to increase rainfall in Horsham or Berri or Wentworth, don't buggerise about building a lake, build a mountain. But you'll have to make it a bloody big one - I've never seen figures on this, but I imagine something in the rough order of two or three thousand metres high and a few hundred kilometres long would do the trick. Mind you, you had better be prepared to deal with the opposition of absolutely everyone who lives on the other sde of your mountain range, because they will get practically no rain at all in the lee of your artifical range.
Actually, thinking about it, it might need to be more like 5000 metres high: the Flinders Ranges and the Macdonnells, after all, are hardly threatening to get themselves listed among the highest rainfall areas of the continent.