If what you want a gut-feeling of what these equations mean, I can share mine.
Equation (2) is a consequence of the two others. Take the time-derivative of (3), remembering that K and $\lambda$ are constants and $H=\dot a/a$, combine this derivative with (1) and (3) and you get (2). So no gut-feeling for that one, just trivial algebra.
Now gut-feeling of (1):
The mass volumic density can change in two ways: either by changing the volume for the same amount of mass (term $-3H\rho$ ) or creating or losing energy (which is the same as mass, of course) because of positive or negative work of the pressure (term $-3HP$). OK, you might not be really convinced, but think it over. If you ignore pressure, then the mass in a comoving volume is strictly constant, and (1) for zero pressure just tells you that the mass volumic density behaves like the inverse cube of $a$.
Finally, equation (3).
Multiply by $a^2$. Then the right-hand-side is $\dot a^2$, kinetic energy per unit mass. Second term, if we had $\rho a^3$ that would be total mass. But we just have $\rho a^2$, so this is mass $M$ divided by radius $a$. $(8\pi GM/3)/a$ opposite of the gravitational potential energy per unit mass. Now $K$ is a constant, total energy. For $\lambda=0$ you have kinetic energy as the total energy minus (negative) potential energy, An intuitive feeling for $\lambda a^2$.... that's another story.
I hope that helps... This is just gut-feeling, it would need a lot of work to make is precise. But his is the way I look at them.