This is bugging me a bit - why would α decay increase the charge?

What is changing is the net charge, which refers to an excess of negative or positive charge. A neutral atom has the same number of protons as it does electrons, and so its net charge is 0. In a decay process an atom will emit, create, and/or annihilate different types of particles directly as part of the decay event, and immediately after the decay there can also be ionization events in which the excited daughter atom and even atoms around it can eject electrons (for example, via the Auger effect, emission of secondary electrons via impact ionization, and other processess).