What chemical reactions take place when mixing baking soda with citric acid and epsom salt and putting it into a microwave oven?
The solid phase reaction of baking soda and citric acid to form monosodium citrate, water and $\ce{CO2}$ takes place very slowly, so you don't get the visible foaming and bubbling that you do when water is added, or if you were to use baking soda with an acidic juice or vinegar (~95% water).  

The molecular formula for Epsom salt is $\ce{MgSO4\dot\ 7H2O}$, meaning it's actually more than 50% water by weight. At $\mathrm{150^oC}$ it starts giving up some of those waters of hydration. Although water is a gas at that temperature, enough of the water is able to wet the baking soda and citric acid to get that reaction going, which then releases more water:  

$$\ce{NaHCO3 + C6H8O7 → NaC6H7O7 + CO2 ^ + H2O}$$  

Once the above reaction is initiated by the release of enough water from the Epsom salt, the additional water the reaction releases makes it somewhat self-sustaining and the $\ce{CO2}$ provides the foam and bubble show. The fact that additional reaction was observed after adding more water just shows that there was unreacted baking soda and citric acid remaining in the solid "crust".