Why are only two of my anchor bolts rusting like crazy?

That's galvanic corrosion almost certainly. You've got a coppery plate and steel bolts sitting in the same wet enviornment, and when rainwater bridges the two, the steel becomes the sacrificial metal and rusts out fast. Copper is the noble metal in that pair, so the bolts are basically dissolving to protect it. Same thing happens with copper flashing on a roof eating the nails below it.

As for why only two corners, my bet is contact and coating. If those two bolts have thin or damaged zinc plating, or they're actually bearing on bare steel while the other two sit on paint, primer, or a burr of plating, they'll go first. Another possibility is a mixed batch. Hardware bins get shuffled all the time and two of your four coud be a diferent coating run. Scratch one of the good bolts somewhere hidden and compare it to a rusty one. If the good ones stay shiny under the scratch and the rusty ones went brown immediatly, you've got your answer. Moisture wicking thru the concrete differently at those two locations is possable too, but the mixed hardware theory is easier to check.

For the stains on the concrete, don't reach for muriatic acid. A rust stain remover with oxalic acid in it is what you want, or even a paste of powdered oxalic acid and water. Scrub with a stiff brush, let it sit, rinse well. Test a small spot first since some concrete finishes can lighten. Get on it soon because the longer the iron sits in the pores the harder it is to pull out. On the bolts, I'd replace all four rather than just the dead pair. If two allready failed in a month, the other two are just running behind, and you don't want mismatched hardware where half is new and half is half-eaten. Since the plate is coppery, stainless anchors are the safe call because stainless won't sacrifice itself to copper. But if the plate is actually weathering steel or plated steel underneath that color, stainless can speed up corrosion on the plate side instead. Isolating with nylon or plastic washers and sleeves between bolt and plate is the belt-and-suspenders fix either way, and it's cheap. Pulling working anchors is annoying but four of the same thing installed the same way beats playing whack-a-mole next spring.

Before you buy anything, hit that plate with a magnet. I chased a similar mystery on a mailbox post and the fancy coppery finish was just paint over plain steel, which changed what hardware made sense. On the anchor pulling question, I've had decent luck not pulling them at all. I cleaned the rust off with a wire wheel, hit the threads with some cold galvanizing spray, and put stainless nuts and washers on top. Three winters now and no new rust. If the good bolts on yours are solid, that might buy you time while you figure out the plate situation.

Had almost the same thing on a fence gate post a couple years back. Only the two bolts nearest my sprinkler head rusted while the far side stayed clean, so it really can come down to where water lands, not just where it pools. Took me a while to figure that out becuase I was looking at the concrete, not up at what sprays over it. Do you have anything nearby that drips or sprays on that pad, like an gutter edge or an irrigation line? Might explain why two corners took the beating and the others didn't.

In my day we used to smear a dab of grease or even tallow on the bolt shank and under the head before setting anchors in concrete, keeps the water from wicking down the threads. Worth a thought here because I bet the two rusty ones are the ones you torqued down hardest. Cranking an anchor bolt crushes the plating right at the washer and you've opened bare steel to the plate without knowing it. I've done it myself plenty. Next time snug them even and check with a wrench, not by feel. Also try a magnet on the rusty pair versus the clean pair before you buy anything.

I had a pair of anchor bolts do this on a lamp base years ago and it turned out the concrete was still curing. Fresh pads wick moisture up around the bolts for a long time and it comes out wherever the mix is denser or wetter, which can absolutely be two corners and not the whole pad. Mine quit rusting once the pad finished drying and I sealed around the bolt bases with clear silicone. Might be worth touching the rusty spots, if the concrete feels damp or darker there on a dry day that's your moisture path and new bolts will just do the same thing.

Update: mystery solved. The magnet trick from the mailbox post reply was the winner - the plate is just steel with a coppery coating, so it was plain steel on steel all along, not anything fancy. The two rusty corners are the ones closest to where the downspout dumps, so water direction explains the pattern. Scrubbed the stains with diluted vinegar, rinsed well, and swapped all four for galvanized bolts with some grease under the heads. Looking clean so far.