IIRC, the Maharal, in Beer Hagoleh ch. 4, explains that the Leviathan is a symbol for the world's great materialism (as it's an enormous mass of flesh-life), and 'playing' is something that may be a source of enjoyment without fulfilling a need. Thus, God is described as playing with earthly materialism, because He doesn't need anything from His creations. There are other recorded explanations as well: a quick internet search showed me that this passage is discussed by R. Tzadok Hakohen (scroll down) and R. Yonasan Eibshutz in Yaaros Dvash 1:7
If I may offer my own "פשט על דרך אגדה", based (somewhat) on ideas that I've heard from R. Moshe Shapiro: the Leviathan is often used by Chazal as a reference to the reward of the righteous in the future (see Bava Basra 74b, and the Targum Yonasan to the verse quoted, Tehillim 104:26). Thus, God's "daily" schedule seems to, at least loosely, parallel the efforts of man as well, just on his own field - we are supposed to study the Torah, use the Torah's yardstick as a guide for how we/the world should operate (parallel to judgement), and implement the Torah's values in order to improve the world and the lives of those who inhabit it. After all the 'work', there is 'play'; in the future, the person who has done what he is supposed to do will be rewarded with being able to enjoy 'the meat of the Leviathan'