A simple explanation is that God restricted Jews from 'work' that affects the world in certain ways on Shabbat in order that they should rest. For example, no building, or lighting fires. However, they can benefit from pre-existing fires or buildings.

Pressing the elevator button is akin to one of these forbidden effects (which one depends on a broad discussion about electricity on Shabbat), going against the injunction to rest. Walking in to the elevator and it moving automatically is more similar to benefitting from an existing fire.

While it may seem strange that pressing the elevator button is considered 'work', the answer to that is that the definition of work is directly related to the laws and not necessarily some external metric like effort. This means that the spirit and letter end up very closely related, but I think the distinction still exists.