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"Travelling at 20% of the speed of light, they would take about 20 years to [reach] Proxima Centauri."

I find those 20% to be an unimaginable relative speed already. Can we do that?!

Not yet, but the article quoted explains (rather optimistically, imho) that it would be doable for very small probes, providing we find sufficiently powerful lasers and targetting mechanisms. They even budgeted a prototype but other people seem to think that the budget should be increased by a factor of 1000. But yeah, technically, it's possible for non manned projectiles.