The one way for this to work and maintain a balance for a long period of time is for a species to have a limit on reproduction or the time it takes to reproduce. Today, life hangs in equilibrium because of limits due to food supply. If there is more food a larger population is able to survive, and this larger population eats the food. There will then be less food, and the animals will die off. To avoid this cyclical pattern, many species reproduce only to the extent that the environment can support them.
For this planet to work, humans (or whatever species it may be) would have to have a more or less stagnant population hovering at the level that the food could support them. If this was achieved, life would stay in balance with very little evolution, change, or adapting, until outside factors influenced the world (new species of predators, climate change, or a disease that removes their food source). This would probably wreak havoc on the unchanging species, possibly even driving them to extinction.
Alternatively, if reproduction wasn't stifled somehow many issues would break out. There would be fights for control of food sources and land, forcing people to adapt and cope with their new environments. In the end it would look similar to the planet now.
Basically there are two main ways it could go. Low prioritizing of reproduction would result in stagnation in the species and any minor change would threaten to drive the species to extinction (think pandas). Alternatively, a fluctuating species population and having reproduction as a priority would look much like the world does now with species fighting amongst themselves and against other species for control.
To directly answer the question, there would be no need to evolve as there would be no challenges facing us. We would only use as much energy as necessary, and we would probably spend our time collecting food. Unless the food source grows at the same rate as the human population we would not be able to advance much and form bigger communities. Overall, life would be...boring.
From a utopian point of view, food really isn't a limiting factor. We have large amounts of food, and as a result our population can grow, but this creates pollution and waste. We could potentially push towards a utopia now if we strictly limited the reproduction rates of humanity and focused on renewable sources of food and energy, but this is a separate topic.