Why does it seem as if big vehicles "attract" mine when I drive close to them?

No, not at all. It doesn't have anything to do with gravity. The masses of the cars are so very small that you can't even think of the gravity between them. There is gravity, but it is negligible.

And one person answered with pressure. It's not that, either. Yeah, if you put two pieces of paper against each other and blow in the middle of them, the papers look like they attracting to each other, but the thing is, the air pressure in the middle is removed and only the pressure of outside is applying force to the papers. However, in our case, the air pressure is not that much to push two cars! If you blew between two pieces of metal, will they bend toward each other? Of course not!

But your answer – it's probably mental. Maybe the fear of hitting to that car or anything else force you to think that you are getting closer to the car but it's not like that. Actually, I've read an article some years ago about when you are sitting in a height and the feeling that you want to push the other person or yourself down, even if you don't want to. But anything it is, it doesn't have to do with a physical phenomenon.

It all has to do with velocities and pressure between the two vehicles. So, in principle, it has to do with Bernoulli.

The two vehicles drag air with them, so between them you will have an area of higher fluid velocity than the area to the left and right of each vehicle respectively. To explain more plainly, between the two vehicles you will have dragged air from the two of them while to the left and right of them you will have dragged air from only one of them. So you have higher fluid velocity between them.

So, by applying Bernoulli's principle you find that between them you have lower pressure (high speed) and outside of them you will have higher pressure (lower velocity), so they get pushed towards the area with lower pressure, which is closer to each other.

When a vehicle moves through a gas or a liquid, it displaces the medium through which it moves, and gas or liquid must rush in to fill the void of displacement. It's exactly like the bow wave of a ship which must fill the area immediately behind the ship. This creates suction past the mid point and toward the stern of the ship. I'm certain that you feel drawn toward a larger vehicle only after it has begun to pass you, or just as you begin to pass its midpoint.

Ships are designed with overhanging sterns so that they will not be swamped by the bow wave as it is sucked into the propellers behind the ship. Incidentally, many clipper ships which disappeared without a trace while racing through the "roaring 40s" were not designed to accommodate the huge bow wave displacements they created at speeds far beyond what sailing ships previously had attained. With sterns incapable of riding atop the displaced water, the clippers may have become submarines, and sailed beneath the bow wave, which tore off cargo hatches and swamped the ship in short order!

When you are drawn toward a larger vehicle, it is because you are caught in the air rushing into the void created by the larger vehicle's motion.

There is a great amount of energy involved here for sure. All moving objects meet resistance and the more velocity the more resistance. For objects moving fast there is aerodynamic designs and properties added so it experiences less turbulent force pressing back on the object. Drafting in race car driving is a useful example of this. The vehicle in front is breaking through the air while the car in behind rides inside the draft or slipstream. Inside the slipstream it will take much less energy to produce the same speeds. So you are not getting pulled in by the vehicle ahead its that under the same power conditions you are meeting much less drag or resistance so you will for sure have to dial back your throttle. The front vehicle breaks the turbulence for the vehicle behind. Read about aerodynamics.

More or less i would agree with the previous answers:

The fast moving big car of your example ,displaces the surrounding air creating turbulence and low air pressure areas , that drags your car close to the other.

In addition , i would like to point out that this is a very well known phenomenon in aerodynamics called 'drafting' or 'slipstreaming'.

It's a very popular technique that car racers use , in order to gain advantage during a race.

More details you can find here: https://en.wikipedia.org/wiki/Drafting_(aerodynamics)