Why do endothermic reactions happen?

There are two possible reasons:

1. because the reaction produce products with higher degree of disorder (e.g. liquid < solutions < gaseous substances, are more disordered than solids) and/or in those cases in which the number of moles of products is higher than the number of moles of reactants (example: decomposition reactions).

2. because the sistem is open, i.e. some product is physically and irreversibly subtracted from the reacting system (e.g. formatin of precipitates, complexes, consecutive reactions where the equilibrium is not reached, as in living systems, etc.)

About point 1. it is worth to know that the tendency to form most stable (energetically) systems, as occurs in exothermic reactions, as measured by a negative enthalpy variation, is not the only driving force of spontaneity of chemical reactions. Another important driving force is the tendency to produce more disordered systems, where the increase of disorder, or increasing probability, is measured by the Entropy variation multiplied by T (T = absolute temperature). For spontaneous-endothermic reactions the Entropy term prevails over the Enthalpy term.

It is easier to understand this circumstance with some common physical transformations, as evaporation in a close bottle: a drop of alcohol or ether evaporates spontaneously even if the evaporation subtracts energy (is endothermic), because the molecules in the vapour phase have much more entropy (disorder) of the same molecules in the liquid phase.
Dissolving common salts in water, as KCl, is often a spontaneous & endothermic process, because the solution is much more disordered than the crystal + separate water, thus driving the process notwithstanding the hydration energy (from ion-water bonds) is lower than the ionic lattice energy (the energy that is consumed to separate the crystal ions) making the whole process endothermal.

About point 2. the Le Chatelier law of equilibrium, states that subtracting a product from the equilibrium state, the system reacts to reproduce a new equilibrium condition, and this implies that new reactants are transformed to restore the subtracted product. This recovery of equilibrium condition after unbalancing occurs independently by the exothermal or endothermal character of the reaction. So, in a open system in which one of the products of the equilibrium continuously excapes, the reaction will go ahead continously until one of the reactants ends, even if the reaction is endothermal and if the entropy term is disadvantageous.

Sometimes the reasons 1 & 2 are combined, when, for example, the reaction produces a gas in an open vessel, as in this very famous and amazing reaction, in which gaseous ammonia, a liquid solution, and more moles of products are produced:

I hope you have got a complete understanding and familiarization with endothermic reactions-transformations.