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"As you can see, we don't get a whole number, because in FOIL, we have middle terms giving us the $2\sqrt6$ part.

"It's different from a fraction, because if you look at the sum $\frac12 + \frac13$, there is a common demonimator $6$ you can multiply by that makes it the sum of two whole numbers: another whole number. No cross terms arise, because there's no FOIL going on."

"As you can see, we don't get a whole number, because in FOIL*, we have middle terms giving us the $2\sqrt6$ part.

"It's different from a fraction, because if you look at the sum $\frac12 + \frac13$, there is a common demonimator $6$ you can multiply by that makes it the sum of two whole numbers: another whole number. No cross terms arise, because there's no FOIL going on."

(* FOIL = distributive rule applied to binomials; mnemonic for "First terms, Outside terms, Inside terms, Last terms")

Also instructive are the cases where it does work. For example, $\sqrt2 + \sqrt8 = \sqrt{18}$. You can "FOIL" it out and see why.

Second edit:

One more run at this, just to see how succinctly I can get the main point.

"To see what something is the square root of, square it:

$$(\sqrt2 + \sqrt3)^2 = 2 + 2\sqrt6 + 3 = 5 + 2\sqrt6$$

"As you can see, we don't get a whole number, because in FOIL, we have middle terms giving us the $2\sqrt6$ part.

"It's different from a fraction, because if you look at the sum $\frac12 + \frac13$, there is a common demonimator $6$ you can multiply by that makes it the sum of two whole numbers: another whole number. No cross terms arise, because there's no FOIL going on."