Why do some mobile devices require a minimum battery charge level before they can be powered up?
When a phone powers up, its boot process is power-consuming. More, during the boot and attempting to connect to network, it will use its radio, which will consume sizable peak currents (0.5-1A). This burst current might be unavailable for general chargers from the wall. 

Now, in accord with Li-Ion safety regulations, all Li-Ion cells must be protected, including over-discharge. This is an independent function of the stock primary cell. If the battery senses its output voltage below 2.5-2.7 V, it will automatically shut itself down, disconnect. If you try to boot a phone on a weak battery, the voltage might droop, and the battery might surprise the operating system with self disconnect, and corruption to boot/system files may occur, and the phone might brick out. Therefore, the battery must have enough "juice" to hold the boot process and sustain short radio bursts. That's why the phone's bootstrap processor waits for proper charging status from the system battery gauge before booting full-rate.

When a discharged cell begins charging, the over-discharge protection turns off, and the internal phone charger will see certain voltage on the cell. It could be just ~2 V or something, still too low for normal operations. In this situation the charger usually enters so-called "pre-charge" mode, charging the cell slowly at 20-50-100 mA rate. This slow-current stage is necessary to revive internal chemistry and prevent cell gassing/bloating. So it might take some time when the cell voltage (Charge Status) gets up to the minimum operating voltage (usually ~3V), when the charger can engage full-scale (fast) charging mode, and the battery has enough charge to power radio bursts without sagging below the battery over-discharge threshold and self-disconnect. All these precautions are necessary to ensure safe and reliable phone functionality.