Intuitive explanation for how could there be "more" irrational numbers than rational?

I think the most intuitive explanation I have heard is to considering writing down a rational number in decimal form. This means that either it is a repeating decimal or a terminating decimal, for example $2.3737\overline{37}$ or $0.42$, which we will write as $0.4200\bar{0}$. Now, consider the probability of randomly writing down a number. So you have ten options every time you go to place a digit down. How likely is it that you will just "happen" to get a repeating decimal or a decimal where you only have zeros after a certain point? Very unlikely. Well those unlikely cases are the rational numbers and the "likely" ones are the irrational.

There is always a rational between two irrationals, and always an irrational between two rationals, so it seems like it should be split pretty evenly.

That would be true if there was always exactly one rational between two irrationals, and exactly one irrational between two rationals, but that is obviously not the case.

In fact there are more irrationals between any two (different) rationals than there are rationals between any two irrationals -- even though neither set can be empty one is still always larger than the other.

And yes, this really becomes less and less intuitive the more you think about it -- but it seems to be the only reasonable way mathematics can fit together nevertheless.

The wording in the question, "it seems like it should be split pretty evenly", is quite appropriate. I think the lesson to be learned here is that sometimes when our intuition says something "seems like" it should be true, a more careful analysis shows that intuition was wrong. This happens quite often in mathematics, for example space-filling curves, cyclic voting paradoxes, and measure-concentration in high dimensions. It also happens in other situations, for example [skip the next paragraph if you want only mathematics]:

I've seen a video of a play in an (American) football game where a player, carrying the ball forward, throws it back, over his shoulder, to a teammate running behind him. The referee ruled that this was a forward pass. Intuition says that throwing the ball back over your shoulder is not "forward". But in fact, as the video shows, the teammate caught the ball at a location further forward than where the first player threw it. If you're running forward with speed $v$ and you throw the ball "backward" (relative to yourself) with speed $w<v$, then the ball is still moving forward (relative to the ground) with speed $w-v$. So the referee was quite right.

In my opinion, the possibility of contradicting intuition is one of the great benefits of mathematical reasoning. Mathematics doesn't merely confirm what we naturally know but sometimes corrects what we think we know.

Intuition is a wonderful thing. It's fast and usually gives good results, so it's very valuable when we don't have the time or energy or ability for a more thoughtful analysis. But we should bear in mind that intuition is not infallible and that more careful thought can provide new insights and corrections.