It's because using air drag to slow you down saves enormously on payload fuel mass.

If you are coming down near Earth from a very high position, you will sweep by the Earth going way too fast to stay in low earth orbit. You'll just fling past Earth and rise up high again. So if you want to dock with something in low earth orbit, like the ISS, you have to decelerate a lot. That excess speed has to be removed to slow you down into a relatively circular orbit at low altitude.

And if you are going to do that decelerating in space, not touching the atmosphere, you have to provide all the deceleration yourself. With your own fuel.

But if you pass really low to Earth, so you are scraping the atmosphere, then the atmospheric drag can provide all the deceleration you need without you having to spend fuel doing it yourself. Granted, that means you need protection from the heating effects of the pressure shock, but if you can handle that heat you gain the advantage of needing no propulsion to decelerate with - that means not needing a powerful engine and not needing fuel for that powerful engine.

Merely having a very small bit of fuel for a little steering engine that can merely slightly deflect your path is sufficient. All you have to do is slightly bend your path long before you get to Earth, to ensure you hit the atmosphere just right. That's more of a computing challenge than a delta-V challenge.

And you don't even need to put that computing power on the probe itself. You can have the probe have just enough smarts to obey any maneuver command you send it, and then use more powerful computers on Earth to calculate the needed maneuver to beam to the probe. (This is basically how they did this sort of thing in the 70's when the math required room-sized computers.)

I didn't cover it yet, but if you try to do your own self-propelled deceleration to turn a high-speed Earth flyby into a low Earth orbit, it's not sufficient to have a little bitty weak steering engine. You need to ensure that engine has a significant level of thrust because it doesn't just need to spend a lot of delta-V, it has to spend it fast. If it's one of these super-efficient but also super slow engine designs, such as a weak little ion engine, the probe won't slow down fast enough and will slingshot past the Earth back up to a high altitude before it has caused enough delta V to do much.

tl;dr - Needing a heat shielding arrangement for the bit where the probe hits atmosphere hard does add a little bit of mass, but not nearly as much mass as it would take to have a high-thrust engine and the fuel to run it so you can provide that deceleration yourself.

And since we're talking about the mass at the very end of the mission, in the final payload that comes home, that's mass you have to carry at the tip of the rocket through all the stages along the way. The very last stage of the mission is the most important place to save on mass. 1 kg more mass in the final stage can mean more tonnes of mass added across the other stages that come before it.