How does aluminium react with bases to form aluminates?
There's no positive charge on Al, yet it reacts with hydroxide groups floating around.


Your observation is correct. In the beginning, aluminium in its elemental state has the oxidation state 0. Apparently, that changes in the course of the reaction:
$$\ce{Al &lt;=&gt; Al^3+ + 3e-}$$


  Where will the electron from a non-ionic aluminium atom go?


Again, your observation is correct. If one species is oxidized, another one must get reduced. In this case, protons from the water are reduced to form hydrogen gas:

$$\ce{ 2H2O + 2e- &lt;=&gt; 2OH- + H2 ^}$$

If you combine both half reactions and balance the stoichiometry, you end up with the initial equation for the whole redox reaction.