Monitors can show only RGB models, which has different values on different settings of the monitor. CEIL LAB was designed to be device independent, however I have a confusion that essentially the monitor is going to convert a given LAB value to RGB, then how can we say CEIL LAB is device independent ?
No. CIELAB images are just as subject to different displays as RGB images. As you note CIELAB is converted to RGB in order to be displayed since displays do not use CIELAB. However, if displays are calibrated to the same white point and profiled then they will display images consistently so long as the images are within the display's gamut.
And CIELAB is not device independent any more than Adobe RGB (1998) or sRGB is. In fact it's more dependent. Technically the two RGB colorspaces require white to be D65 while ProPhoto RGB specifies D50, even though in practice people often set their monitor's white point to other values.
CIELAB, on the other hand doesn't have a specified white point. So a LAB value is technically meaningless unless accompanied by a white point (in CIE XYZ). So it is simply scaled to whatever the display is set to.
For purposes of printing, the ICC specifies that LAB is referenced to D50 white point and further restricts the a* and b* range to +- 127/128.
Even though CIELAB has a much larger gamut than most RGB colorspaces there are surprises. For instance Adobe RGB (0,255,0) is slightly outside this ICC restricted LAB gamut. It has an a* of -129.1 and is clipped to -128.