Hi Rahul,

Using Epilog for the axis labels is not a good idea. The labels do not follow the axes as the plot is rotated. Use AxesLabel instead.

AxesLabel -> (Style[#, 14, Black, Bold] & /@ {"\[Xi]", "K", "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(r\)]\)"})

To make the plot legend bold

PlotLegends -> {Style["Real frequency", Bold]}

To change the tick style

TicksStyle -> Directive[Bold, Black]

Result

Plot3D[Re[x], {xi, 0.1, 1}, {k, 0.1, 1}, 
 BoundaryStyle -> Directive[Blue, Thickness[0.007]],
 PlotLegends -> {Style["Real frequency", Bold]},
 AxesLabel -> (Style[#, Black, 14, Bold] & /@ {"\[Xi]", "K", 
     "\!\(\*SubscriptBox[\(\[CapitalOmega]\), \(r\)]\)"}),
 Mesh -> {{0}},
 BoxRatios -> {1, 1, 0.4},
 BoxStyle -> Directive[Black, Thickness@0.006],
 TicksStyle -> Directive[Bold, Black]]