Why is the Big Bang theory being largely accepted and why are the steady state and quasi steady state not being accepted?
Sriram
Because:
a) the former ("the Big Bang theory") is consistent with a huge number of experimental and observational results, within its stated domain of applicability; and
b) the latter ("the steady state and quasi steady state") are inconsistent with a great many such results.
A widely cited paper outlining a) is http://arxiv.org/abs/astro-ph/0603449" , by Spergel et al. (the link takes you to the arXiv preprint).
At an extreme level of summary, it is the quantitative details of the CMB (cosmic microwave background) observations which b) are most strongly inconsistent with.