Literally: again – three – again – four
Alternately: Repeatedly; over and over; time and time again. It seems that this phrase is often used while chewing someone out, to emphasize how many times they were given a warning or piece of advice (which they then ignored, to unfortunate effect).
Notes: Supposedly the human brain, while capable of some pretty advanced counting and other mathematical tricks, tends to weight triples with indefinite extension: something that happens once is noteworthy, twice is a coincidence, and three times means it’s the rule. This compound plays on that by combining the power of three with the power of escalation and the power of repetition to really drive home the point. Note that 再三 can also be used on its own with the same meaning.
Apparently this simple phrase comes to us from the Qing-era Chinese novel, Hongloumeng (Japanese 『紅楼夢』 = Kouroumu), a.k.a. Dream of the Red Chamber or The Story of the Stone.