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I am trying to understand Bayes Factor (BF). I believe they are like likelihood ratio of 2 hypotheses. So if BF is 5, it means H1 is 5 times more likely than H0. And value of 3-10 indicates moderate evidence, while >10 indicates strong evidence.

However, for P-value, traditionally 0.05 is taken as cut-off. At this P value, H1/H0 likelihood should be 95/5 or 19.

So why a cut-off of >3 is taken for BF while a cut-off of >19 is taken for P values? These values are not anywhere close either.

I may be missing something very basic since I am a beginner in this area.

A few things:

The BF gives you evidence in favor of a hypothesis, while a frequentist hypothesis test gives you evidence against a (null) hypothesis. So it's kind of "apples to oranges."

These two procedures, despite the difference in interpretations, may lead to different decisions. For example, a BF might reject while a frequentist hypothesis test doesn't, or vice versa. This problem is often referred to as the Jeffreys-Lindley's paradox. There have been many posts on this site about this; see e.g. here, and here.

"At this P value, H1/H0 likelihood should be 95/5 or 19." No, this isn't true because, roughly $p(y mid H_1) neq 1- p(y mid H_0)$. Computing a p-value and performing a frequentist test, at a minimum, does not require you to have any idea about $p(y mid H_1)$. Also, p-values are often integrals/sums of densities/pmfs, while a BF doesn't integrate over the data sample space.

The Bayes factor $B_{01}$ can be turned into a probability under equal weights as
$$P_{01}=frac{1}{1+frac{1}{large B_{01}}}$$but this does not make them comparable with a $p$-value since

1. 
   $P_{01}$ is a probability in the parameter space, not in the sampling space


2. its value and range depend on the choice of the prior measure, they are thus relative rather than absolute


3. both $B_{01}$ and $P_{01}$ contain a penalty for complexity (Occam's razor) by integrating out over the parameter space

If you wish to consider a Bayesian equivalent to the $p$-value, the posterior predictive $p$-value (Meng, 1994) should be investigated
$$Q_{01}=mathbb P(B_{01}(X)le B_{01}(x^text{obs}))$$
where $x^text{obs}$ denotes the observation and $X$ is distributed from the posterior predictive
$$Xsim int_Theta f(x|theta) pi(theta|x^text{obs}),text{d}theta$$
but this does not imply that the same "default" criteria for rejection and significance should apply to this object.