The basic idea is that things in nature like to have a low amount of potential energy. For example, a ball that is high in the air has a high amount of gravitational potential energy. That ball (unless supported) will fall, releasing that potential energy and moving to a state of lower potential energy. Chemicals behave in the same way. Enthalpy is a measure of chemical potential energy. An exothermic reaction releases potential energy so that the products have a lower potential energy than the reactants. Conversely, an endothermic reaction raises the potential energy of the products relative to the reactants.
So, to answer your question, highly exothermic reactions are likely to be spontaneous because going from a state of high potential energy to a state of low potential energy is favorable.
However, potential energy is not the only factor involved in determining whether a reaction is spontaneous or not. In addition to minimizing potential energy, nature also tends toward maximizing the entropy (disorder) of the system. Endothermic processes can and do happen because the end result increases entropy. For example, when you dissolve certain salts in water, the solution becomes colder, indicating that the dissolution of the salts was endothermic. However, the dissolution also breaks apart a highly ordered crystal lattice, causing a large increase in entropy. This increase in entropy dirves the dissolution.
You can use the following equation to determine whether a reaction will occur sponaneously:
ΔG = ΔH - TΔS
where ΔH is the change in enthalpy, T is the temperature, ΔS is the change in entropy, and ΔG is the change in a quantitiy called Gibbs Free Energy. If ΔG < 0 then the process is spontaneous. If ΔG > 0 then the process is not spontaneous (unless work is done on the system by an outside force).
[note: minimizing the potential energy of the system and maximizing the entropy actually turn out to do the same thing (maximize the number of ways that you can distribute energy throughout the system), but you learn this in very advanced physical chemistry courses].