It appears to be the slot that is called when the model selector is updated.  I'd recommend reading or skimming http://doc.qt.digia.com/stable/model-view-programming.html for more context.  As to what it does that is self-explanatory from the code.  It enables the buttons based on what is selected and looks like it enables them when it's not a root node.

Like I commented above, it's bad practice to name a slot on what it is connected to.  It's better to name it what it _does_ to the target object.  It might be better to rename this slot

    void UAVObjectBrowserWidget::updateButtonsEnabled(const QModelIndex &current, const QModelIndex &previous)

to be more explicit since that is all it does.