There are two claims here, one about increasing reliance on electrical energy, the other about the information age. You can operationalize both and look for data that supports the claim. The first claim can be operationalized as electricity consumption (worldwide or in the OECD), over time. There's data published, for example, by the international energy agency. The second claim can be operationalized as growth in petaflops, available computers or smartphones per household, people having internet access, you name it.
Another, rhetorically nicer but less scientific, at least less scientific-sounding, approach to 'proving' the obvious is to use telling and vivid examples:
When a large-scale electrical storm went through northern Virginia on
29 June, electrical load in one of more than 10 Amazon data centers in
the region failed to successfully transfer to generator backup.
Uninterruptible power supply (UPS) serving that particular circuit
eventually ran out of battery power and servers began losing power
around 8pm PDT. This caused service interruptions for many online
businesses -- including popular ones like Netflix and Pinterest
(liberally quoted from here).
Also, there is this IEEE paper on worldwide energy needs of information technology.