Why do we have the airgap ? Is it just to store energy? I did find an explanation online, but it was hard for me to understand.
An air gap reduces the effective permeability of the magnetic core therefore, the formula for flux density based on magnetic field strength (H) and \$\mu\$ predicts that B (flux density) drops.
Remember that B = \$\mu\$H and, if \$\mu\$ has dropped for a given H field, then B has also reduced. If B reduces then the core will saturate less at a given current.
But, reducing \$\mu\$ can also increase H so it can seem a bit counter-intuitive. Here's how it pans out for a simple inductor: -
If you reduce \$\mu_r\$ by 50% then inductance halves so you then need to restore this by increasing the turns BUT you only need to increase turns by \$\sqrt2\$ to regain the original inductance.
Now, H is ampere turns per metre and if "turns" have increased by \$\sqrt2\$ then the H field has increased by \$\sqrt2\$. But this isn't a problem because if you go back to the first formula with \$\mu\$ reduced by 2, the B field has dropped to half so, the net difference is that halving the permeability \$\mu\$ means the H field has risen by \$\sqrt2\$ but, the net effect on B is that it reduces by \$\sqrt2\$.
All of this means that, an inductor without an air gap will saturate at a lower current compared to one with an air gap (all other things being equal).