Why the current entering and leaving the battery in an electric circuit needs to be same?
There's the usual, droll electronic laws answer. But I'll take a different perspective that I think gets the point across more deeply. 



Take a capacitor. You know that it can store charge, right?

So, given your thinking process, you might then imagine that the current on one of the terminals can be different from the current at the other terminal of the capacitor. And you might ask, why must it always be that the current goes through the capacitor instead of having current X leaving the left terminal of the capacitor and current Y entering the right terminal of the capacitor, where \$X \ne Y\$. Right? I mean, you would think this could possibly happen?

Not usually. Just possibly?

Or, perhaps, that we might set up two capacitors in series (like your batteries) and where you would imagine that one of them could have a different current in it than the other one. One could "build up charge" so to speak, you might argue.

Yes?

No.

Sometimes, it is very hard for people to realize this. But the force acting between two charges is huge. Not just big. Not humongous. But unimaginably huge. The effect is that matter in nature is essentially neutral. At all times. Ionization does occur. But it is ever only for very small numbers of charges.

Let's take your two batteries. They are separated, let's say, by a distance of \$10\:\text{cm}\$. They are each \$10\:\text{cm}\$ wide, too. So the mean distance between their centers is \$20\:\text{cm}\$. Let's say that the current leaving and entering one of these batteries is \$1\:\text{A}\$ and the current leaving and entering the other one (in series, as you say) is \$2\:\text{A}\$. Then it must be the case that a charge differential is building up between them, based up the missing \$1\:\text{A}\$ that is accumulating into one of these batteries.

(Keep in mind we are assuming magic batteries here that can actually manage to keep and hold a lot of charge.)

How long might you wait until the force acting between these two batteries is 2000 lbs, or one ton!! Well, it's \$20\:\text{cm}\cdot\sqrt{\frac{8900\:\text{N}}{k_e}}\approx 200\:\mu\text{C}\$. At the differential rate of \$1\:\text{A}\$, this would take about \$200\:\mu\text{s}\$. In that short time, assuming magic batteries here, you'd already have an incredible force acting between those batteries.

In reality, that of course does not happen.

There can be moments when things are not exactly in balance. But these are for very short moments while charges redistribute.

The universe needs pretty much everything in it to be relatively neutral.

Gravity on the other hand is pathetically weak. You can accumulate a lot of mass before any serious force occurs. But with electric charges?? WoW! No way. Everything stays pretty much neutral pretty much all the time.

At least, on Earth.



So why do all the currents summing into a node have to equal all the currents exiting the same node?? Because if that didn't happen, it would take no time at all for the circuit to be literally attracting nearby planets towards us.. or perhaps repelling the Earth from the sun at a fast pace.

It would be cool, I suppose. That kind of power would be nifty to possess. Rocket ships would be trivial to launch. Just turn on the current for a little bit and whoosh!! Up into space they'd go.

Life would be so much different.

But then we'd probably not be here, either.