Physicists always ask why the cosmological constant is not exactly zero! I would ask here, what if cosmological constant was zero? The universe wouldn't expand and matter would exert gravitational force and shrink the universe into a big crunch! So, why physicists want the constant to be zero then? I must have missed something here!
Can cosmological constant be zero since we see the universe already expanding? How would the universe support life further as some claim?
The universe can expand just fine without a cosmological constant. In fact, it was this fact that made Einstein originally add it to the equations when he was making his first cosmological model: he did not know space-time is expanding, so he used a constant as an allowed but ugly fudge-factor to make it static in his model. Later he felt he had made a mistake and should have trusted the math (in an extra heaping of irony current cosmological measurements do find acceleration best described by having a constant). But expansion can happen without it.
Assuming the universe to be spatially homogeneous and isotropic, and combining this with the Einstein field equations produces the two Friedmann equations $$\frac{\dot{a}(t)}{a(t)} = \frac{8\pi G}{3}\rho - \frac{k}{a^2(t)}+\frac{\Lambda}{3}$$ and $$\frac{\ddot{a}(t)}{a(t)}=-\frac{4\pi G}{3}(\rho+3p)+\frac{\Lambda}{3}$$ where $k=+1,0,-1$ depending on curvature. $\Lambda$ is the cosmological constant.
Note that if we want $\ddot{a}(t)=\dot{a}(t)=0$ (no expansion) and $\Lambda=0$, then the first equation implies $\frac{8\pi G}{3}\rho a^2(t) = k$. This will not work if $k=0, -1$ since the left side is nonzero and positive. The second equation leads to $\rho+3p=0$: for any positive density there has to be negative pressure even if we are just thinking of the contents of the universe as pressure-free dust. So it looks like $\dot{a}(t) \neq 0$... unless one adds a suitable nonzero value of $\Lambda$ to make things stand still.
To add to Anders Sandberg's answer, the Friedmann equations are really the crucial piece of the puzzle here. These equations assume General Relativity, as well as homogeneity and isotropy (i.e. the universe looks the same in every direction + looks the same at every point). Manipulating the Friedmann equations yields a critical density
$$\rho_c = \frac{3H^2}{8\pi G}$$
The big crunch only happens if the matter density of the universe is larger than this. We observe a matter density that's significantly less than this, which means that even if there were no cosmological constant, the universe will keep expanding. Gravity will slow the expansion down, but it'll never slow to the point where the universe reverses and starts contracting.
From the point of view of cosmology we measure a very small but non-zero cosmological constant and want it to be this value we measured.
From the point of view of quantum field theory, however, it is very unnatural that the cosmological constant is non-zero and very small. The question is similar to the matter-antimatter asymmetry: In the early universe we had a lot of matter and a lot of antimatter, about the same amount of each but not exactly. Then matter and antimatter annihilated but due to the very small difference in amount some matter remained which is all the matter which we observe today. So we could ask: Why were the amounts of matter and antimatter so similar but still not exactly equal?
Similarly, there are hypotheses that the small value of the cosmological constant emerges due to zero point energies of several fields (which individually are very big, rough estimates give over 120 orders of magnitude compared to the value of the cosmological constant) almost cancelling out, but not exactly, leaving a small difference which is the cosmological constant we measure. This is the so called naturalness problem: It seems like an unnatural coincidence that two or several very big values add up to a very small but non-zero value.
