Why there is hydrogen bonding involved in salt bridge formation between glutamic acid and lysine?
An electrostatic interaction is one between a centre of positive charge and a centre of negative charge. In your example, instead of a primary ammonium cation ($\ce{H3\overset{+}{N}R}$) you could also consider the fully methylated quarternary ammonium cation $\ce{Me3\overset{+}{N}R}$ which would still be able to build up an electrostatic interaction with the acid anion. The centre of positive charge is somewhere between the three ammonium hydrogens and the centre of negative charge somewhere between the two carboxylic oxygens.

A hydrogen bond, on the other hand, is a directional interaction of the type $\ce{X-H\bond{....}Y}$, where $\ce{X}$ and $\ce{Y}$ are electronegative elements and $\ce{Y}$ must feature a lone pair. Thus, hydrogen bonds always contain three atoms and only those three. One one of the three ammonium hydrogens can form a hydrogen bond with only one of the two oxygen atoms.