There is very simple chemistry behind this and nothing more. The magician cuts the lemon into half and a places a tiny piece of sodium between the lemon. 

This Sodium reacts with lemon juice and produces hydrogen. It is an exothermic reaction which produces.  

This hydrogen starts burning in the air in the presence of heat produced which we see as fire.

Through this question we are going to understand a new concept- types of chemical reactions.

Exothermic reactions – Those chemical reactions that result in heat produced. Examples – burning of substances

Reactant → product + energy (in the form of heat)

Endothermic reactions – Chemical reactions that require heat to start. Example – Ice Melting, Photosynthesis

Reactant + energy (as heat) → products